"I turned off the light."
There are quite a few nuance variances but in general the structure is 把+ noun + verb where the noun is the direct subject of the verb.
In a lot of the cases you could rearrange the sentence to verb + noun order to convey the same basic meaning (but with different emphasis), like the sentence here, instead of 我把灯(noun)关(verb)了 you could say 我关灯了. The sentence with 把 put the emphasis on the verb.
Yes, I would think "object". However, it's almost as if the main verb becomes an adjective, e.g. in translation in my other comment that I used to mirror the structure, "I rendered the light closed (i.e. off)".
It literally means something like grasp or take and it's often used to introduce a direct object.
So think of this sentence as "I took the light. (I) turned it off."
Simple and good explanation! A point to note is that it is pronounced ba3 (not sure what the audio pronounces since I usually turn it off).
An easy way of thinking of it, as LeiFeiRalf says, is "I took the light and shut it off", as one sense of "把" is "take hold of", and this is a more colloquial turn of phrase.
Another way of thinking about it is "I rendered the light closed (i.e. off)." This would be stilted in English but it mirrors the grammar of the Chinese more directly.
Is the 把 really necessary? Wondering since the "can I turn the light on? / 可以开灯吗" doesn't have it.