You don’t say “j’ai mange” mais “j’ai mangé”. It is the past participle of “manger”. J’ai manger* amounts to saying “I have do”.

I might not be an expert, but just thinking logically (and translating literally):
J’ai mangé. (I have eaten)
“Mangé” meaning “eaten”. But when you say “J’ai manger” it would literally be translated as “I have eat”, because “manger” is the infinitive.

That’s a very basic grammar concept. The past tense (passé composé) of a verb in -er is être or avoir + the past participle, -é.

• Je suis allé.
• J’ai mangé.
  Etc etc

Note the use of the accent on é. It’s there for a reason, too.

Because the French language has rules!

“Mangé” = “eaten” (= past participle of “eat”) in English. “J’ai” = “I have”, as in either physical possession or “I have ” in the past.

This implies that “J’ai mangé” = “I have eaten”!

The best way of seeing how wrong “J’ai manger” is to consider its best semi-literal translation which is “I must to eat”.

“I must eat” = “Je dois manger” (which also means “I have to eat”) so there is no translation of your phrase which makes sense in English!