Professing themselves to be wise, they became fools the italicized words form a(n): participial phrase infinitive phrase dangling modifier gerund phrase

If the italicized words are professing themselves to be wise, then this is a participial phrase. 
It describes what the pronoun they is doing. 

The answer is the fourth option: it explains the rhythm of the poem.

Rhythm, in terms of poetry, is expressed through stressed and unstressed syllables. In poetry, iamb is defined as a foot or a beat. It consists of an unstressed syllable followed by a stressed syllable. Penta- means five, so if a line of poetry is written in iambic pentameter it would have 5 feet of unstressed and stressed syllables alternating.