We see the spiritual connection between Cloud and Aerith begin in this scene:
Aerith: Cloud, can you hear me?
Cloud: Yeah, I hear you. Sorry for what happened.
Aerith: Don't worry about it.
Cloud: ...I can't help it...
Aerith: Oh... Then, why don't you REALLY worry about it? And let me take care of it yourself. So you don't have a breakdown, okay?
Cloud: What is this place?
Aerith: This forest leads to the City of the Ancients... and is called the Sleeping Forest. It's only a matter of time before Sephiroth uses Meteor. That's why I'm going to protect it. Only a survivor of the Cetra, like me, can do it. This secret is just up here. At least it should be. ...I feel it. It feels like I'm being led by something. Then, I'll be going now. I'll come back when it's all over.
Cloud: Aerith?
--Aerith leaves, but Cloud couldn't move towards her--
Sephiroth: Hmm...... She's thinking of interfering? She will be a difficult one, don't you think? We must stop that girl soon.
Cloud interacts with both Aerith and Sephiroth in the Sleeping Forest and discovers that Aerith is attempting to stop Sephiroth. We also learn that Sephiroth is chasing after Aerith to prevent her from summoning Holy. This scene is why Cloud says he can feel both Aerith and Sephiroth in his soul.
After Cloud and Aerith's spiritual connection in the Sleeping Forest, Cloud says he knows that both Aerith and Sephiroth are in the Forgotten City.
Cloud: "I feel it..."
Cid: "Do you know what TIME it is!?"
(Cloud turns to them.)
Cloud: "Aerith is here.
...and so is Sephiroth."
Cid: "Wa, wa, wait a minute. You serious!?"
Tifa: "But how can you tell?"
Cloud: "...It's not an excuse. I feel it in my soul."
What's interesting is that Cloud says he can hear Aerith's voice from the entrance of the Forgotten City, even though Aerith is in an underground temple:
These
three scenes make it clear that for whatever reason, Cloud can interact with Aerith in a way that no one else can. This is confirmed even further at the end of the game when Aerith reaches down to Cloud from the sky:
Now, although Aerith is dead, in
Maiden of the Planet, there are two lines that suggest Aerith knows she can continue to interact with the living:
"She was told that death was not the demise of life." -Maiden of the Planet
"Aerith knew that death didn't mean to be annihilated." -Maiden of the Planet
These two lines are further supported by these scenes and official quotes:
1. "I tried to create an atmosphere in which she still seems to be by his side - spirit at least." ~Nojima, Reunion Files, pg. 58
2. "As for Cloud... he sees Aerith several times throughout the film. It's not that he sees her because he feels her presence. He sees her because her consciousness lives on inside him." -Nomura
3. "In keeping with the notion that Cloud is in the live action world in the end, we filmed the video for the ending credits in Hawaii. There are fields of flowers on both sides of the road, and the colors - yellow and white - are the same as the flowers in Aerith's church." -Nomura, Reunion Files, pg. 87
4. "She's presented in this movie as Cloud remembers her, and her spirit is still with him - talking to him throughout the film." -Maaya Sakamoto (Aerith's Japanese voice actress), Reunion Files, pg. 58
5. "An answer from the Planet… the Promised Land… I think I can meet her… there." -Cloud Strife
6. "Cloud's friends sent him toward Bahamut like a relay race, and as for the last push---Aerith!" -Final Fantasy VII 10th Anniversary Ultimania, Aerith's profile
7. "The thing which she is unable to hide in her irritation towards Cloud is the fact that he isn't merely dragging the past around, but because that reason might perhaps be related to Aerith.
...
AC: Upon knowing that Cloud had been residing in Aerith's church after leaving the place they had been living in together, her expression becomes complex."
 -Tifa's character profile, 10th Anniversary Ultimania
8. "Since Sephiroth exists, Aerith must exist. There was no doubt about that one." -Nojima, Reunion Files
9.
10.
11. "The voices of two people very dear to him, who are no longer with him. Playfully and kindly, they give him a message: he doesn't belong here yet." -Final Fantasy VII 10th Anniversary Ultimania (Revised Edition) AC Playback
So although Cloud is now free to live his life on Earth with no more guilt or remorse, he will one day reunite with Aerith in the lifestream.
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Here is another post I saw about SE's "love beyond death":
Square-Enix Supports the Idea of Love Beyond Death.
It amuses me how many people will say that Cloud cannot be with Aerith because she is dead and then turn around and say they are fans of Tidus/Yuna from FFX. Tidus and Yuna's love story was directly parallel with that of Cloud and Aerith. Tidus and Yuna's love was based on the concept of loving someone even after they have died. It was also about continuing to love that person and search for a way to be with them again, despite the overwhelming odds. Yuna searched for a way to be with Tidus again (FFX-2) much in the same way Cloud stated that he thought he could "meet" Aerith in the Promise Land at the end of FFVII. Cloud is also portrayed as searching for Aerith and/or meeting Aerith in subsequent games like Kingdom Hearts, Final Fantasy Tactics, and Dissidia Final Fantasy.
Even in FFVII itself, we see other pairings that portray love beyond death. One of these is Vincent and Lucrecia. Years later, on into the Compilation game Dirge of Cerberus, Vincent has not moved on. He is still in love with Lucrecia and often visits a cave where she can be seen, her image preserved in a large crystal. Barret also chose not to move on after the death of his wife. For that matter, neither did Dyne, who later died and was reunited with his wife's spirit in the Lifestream. Even Elmyra, Aerith's adoptive mother, did not move on after the death of her husband.
So, why is it a requirement that Cloud should "move on"? Why is it so pertinent that he cease loving Aerith just because she has died? Vincent, Barret, Dyne, and Elmyra all chose to remain alone, preferring to live with the memory of their loved ones close to their hearts. At least in Cloud's case, he can be with Aerith. She is still with him, even after her death, due to her Cetra heritage. The others I have listed didn't have that luxury... yet they chose to carry on loving their significant others even after they were gone. I think this says a lot for Cloud and Aerith and why they shouldn't be dismissed completely just because one of them has departed the physical plane.
In FFVIII, you see another great example of love beyond death- Laguna and Raine. At the end of the game, the player is shown Laguna standing at Raine's grave, still wearing his wedding ring. He has chosen to continue loving Raine. He has chosen to not move on from her memory.
In FFX-2, Shuyin and Lenne's love endured through 1,000 years of separation. And some people say they find it hard to believe that Cloud can still be in love with Aerith after 2 or 3 years? I believe I have provided more than enough proof that S-E does not consider death to be the "end" of love. Nor do they consider time an obstacle. If Shuyin and Lenne can continue loving one another after 1,000 years, then Cloud continuing to love Aerith for the remainder of his life should not even be questioned. Why? Because S-E says so.
And, finally, in FFXII, Princess Ashe is shown grieving for her lost love, Rasler. Throughout the game, she is shown remembering him many times. Most importantly, though, is the fact that Ashe chooses not to move on. In the end, she remains alone, the memory of Rasler still close to her heart.