It means “not much”. It is used to describe the degree of something that is not great. It makes the negative sentence softer. “あんまり” is derived from “あまり” and it is often used in spoken language.
I may not be able to speak Japanese because I haven’t been studying so much recently.
最近あまり勉強していないから、日本語を喋れないかもしれない。
最近あまり勉強していないから、日本語を喋れないかもしれない。
Ken and Leo are talking about what they have for lunch.
- Ken: Do you want to grab a bite after our tasks are done?
作業が終わったら、ご飯を食べに行かない？ - Leo: Yeah. It‘s not so hot today. I’d like to have spicy food such as Thai curry.
うん。今日はあんまり暑くないから、タイカレーとか、辛い物を食べたい。 - Ken: Sounds nice. I didn't use to like hot food, but as I got older, I came to like it.
いいね。辛い物はあんまり得意じゃなかったけど、大人になるつれて好きになったよ。
Meg went out with Ken last night. It was their first date.
- Amy: How was your date with Ken? Did you enjoy?
ケンとのデートどうだった？楽しかった？ - Meg: Not really. I didn’t know what to talk about because we don’t have so much in common.
あんまり(楽しくなかった)。あまり共通点がないから、何を話せばいいか分からなかった。 - Amy: Did you know that he’s studying Cantonese too? You should talk to him about it!
彼も広東語を勉強しているの知っていた？それについて話しなよ！