You usually learn about いる with other subsidiary verbs such as みる、おく、いく、くる、しまう、…
They all follow the te-form of a verb, are usually written in kana and slightly change the meaning of the verb.
～ている is often translated with the progressive in English
I am eating an apple
You can think of combination of [te-form of a verb] + [subsidiary verb] as a new verb. Conjugation is done on the subsidiary verb. For example
past tense of masu-form of いる = いました
past tense of masu-form of 食べている = 食べていました "I was eating"
past tense of masu-form of 食べる = 食べ ました "I ate"
Literally いる means "to be/exist/stay", so literally, 食べて居る【いる】 means "to eat and stay [that way]".
The fact that this literal meaning is lost makes いる a subsidiary verb. That the literal meaning is lost is also a good reason to write いる without kanji.
(This is the same with other subsidiary verbs. For example, ～てみる is often translated "to try to [verb]". Literally, 食べて見る means "to eat and see [what happens]". This literal meaning is lost and みる as subsidiary verb is written in kana.)
N.B. in ～ている the い is often omitted in colloquial speech/writing, e.g.
食べてた = 食べていた "I was eating"