In poetry, meter refers to a generally regular pattern of stressed and unstressed syllables. I'm sure you've read some sonnets - typically these have a very regular iambic pentameter, meaning, 5 beats per line but a rhythm like a heartbeat.
Anyway, "As I Grew Older" by Langston Hughes does not have a meter. It is free verse - which means it does not have any regular rhyme scheme, meter, nor structure.
If I had to define it, I'd say the rhythm of this poem is dictated by emotion. At the beginning the speaker has a dream and the rhythm of the poem is steady. Then, he hits the wall of opression. Notice how many lines are one or two words only ("Slowly" / "The wall." / "Shadow.
As the speaker gets more excited about breaking free from this opression (at the end) the rhythm speed up - there are more words per line.
Try reading the poem aloud a few times (slowly and with passion) and see how you feel.