What's the grammar of て-form + いて?

The いて I'm referring to is in the last line of the lyrics of a song called Don't you see!:

私をつかまえていて

So apparently つかまえて is the て-form of つかまえる. But why is there a いて at the end?

Answer

いて is the casual imperative of いる (the same as the て-form).

So it's now down to what つかまえている means. The て-form of a verb plus いる can have many different translations (and I'm sure you will find many applicable questions on Stack Enchange). I will attempt to distill it to its most general sense. The て-form plus いる conveys the state that the subject is in after having performed the verb.

The て-form plus いる usually translates to the present progressive or the present perfect in English. The best translation depends on context and the type of verb. You could ask yourself, "What is the subject doing after having initiated the verb?" or "What is the state of the subject after having initiated the verb?"

In the case of some verbs (verbs where the action continues after being initiated like 遊ぶ｛あそぶ｝　play, する do, 寝る｛ねる｝ sleep, 泣く｛なく｝ cry, etc.), the state the subject is in is "still doing the verb". These situations are best translated to the English present progressive or variations having the same meaning (遊んでいる｛あそんでいる｝is playing, している is doing, 寝ている｛ねている｝ keeps sleeping, 泣いている｛ないている｝ continues to cry, etc.)

In the case of other verbs (verbs where the actions are momentary and result in a new state rather than a continuing state like 帰る｛かえる｝ return home, 着く｛つく｝ arrive at, 死ぬ｛しぬ｝ die, etc.), the subject does not continue the verb but rather is currently in the state of "having done the verb". These cases are usually best translated as the present perfect in English 帰っている｛かえっている｝ have returned home, 着いている｛ついている｝ have arrived at, 死んでいる｛しんでいる｝ have died.

So in your case, ask yourself, "What is the current state after the subject grabs hold of (catches, holds, seizes) the object?" The state is that she is "holding the object (or continuing to hold, keeping hold, etc.).

So making that into a command, you have

Keep holding onto me.