"大げさ" is a noun that means "to exaggerate". It is used when describing situations in which someone makes something seem larger, more important, better, or worse than it really is.
"話を盛る" is a phrase that has the same meaning with "大げさ", but it is only used to describe situations in which someone makes a story seem larger, more important, better, or worse than it really is.
You may often hear "盛っている" or "盛った" in everyday conversations.
Ken passed out during work and wake up afterwards.
- Leo: Are you okay?
大丈夫？ - Ken: I'm okay. It seems that I fainted from anemia.
うん。貧血で倒れたみたい。 - Leo: Lie down until an ambulance arrives!
救急車だ車で、横になっていて！ - Ken: What? It's just anemia. You’re blowing it out of proportion.
えっ？ただの貧血だよ。大げさだよ。
Meg and Amy are talking about what they want to have for lunch.
- Meg: Do you want to go to ABC restaurant? Their steak is really good. It's bigger than my face and very thick!
ABCレストランに行かない？あそこのステーキ、すごく美味しい。私の顔より大きくて、超分厚いの！ - Amy: Really?
そうなの？ - Meg: I might be stretching the truth, but it's definitely delicious.
(話を)盛った(or大げさ)かも。でも、本当に美味しいの。
Meg and Amy are gossiping.
- Meg: You know, it seems like Kumi's boyfriend is kind and so cute. He looks like Park Seo-jun!
ねぇ、クミの彼氏って、優しくて、すごく格好良いらしいよ。パクソジュンみたいだって！ - Amy: Are you serious? She was exaggerating a little, right?
本当？(話を)盛っている(or大げさ)でしょ？ - Meg: Could be, but she told me that she doesn't think she'll ever meet anyone like him again.
そうかも。でも、彼みたいな(素敵な)人には二度と出会えないだろう、って言ってたよ。 - Meg: I think she made her story sound much more dramatic than it really was. It's too early to say something like that because they've just started dating.
随分大げさ(or話を盛り過ぎ)だなぁ。付き合い始めたばかりなのに、そんな事を言うなんて早いよ。