As to this phrase burst into the seams, Longman dictionary gives its explanation like this:

if a room or building is bursting at the seams, it is so full of people that hardly anyone else can fit into it.

But in the following conversation, it seems pretty diffirent. Who can tell me the true meaning? And where does this phrase come from? Thanks!

Joe: …one of the reasons I don’t see her that often is because she’s been really busy, uh, because she’s in a new relationship she said.

Kristin: Oh!

Joe: Yeah, you should have seen her. It was like, uh, it was so cute to see her. She was like bursting at the seams she was so happy.

Kristin: [laugh]

Joe: Because I guess the relationship is going really well, so…

The seams are the places that your clothes are sewn together, eg for your shirt, where the sleeves are sewn to the body.

If you put on a lot of weight, your shirt will 'burst (tear open) along the seams'.

The phrase usually means that a place is very, very full.

eg The restaurant was bursting at the seams.

It is not commonly used about a single person, so your example is unusual.

Students: We have free audio pronunciation exercises.
'Bursting at the seams' (not 'into the seams') means something is so full it is nearly bursting/falling apart/exploding. 'Seams' are where pieces of fabric join, so if something (e.g. a bag) is very full, the seams will split and it will 'burst at the seams'.

It's also used metaphorically to mean 'very full' as in the building example you mention. As referring to a person, in the sentence above, it means 'so full of happiness, she is about to burst'.

In this sense, it usually means full of happiness or excitement. But you should always say 'with happiness' or as here 'she was so happy' or the phrase can be confusing - people could think it means that she was so fat, her clothes were bursting at the seams!
Then that makes sense. You said a mouthful. I got it. Thank you!