LITERAL ANALYSIS: SEGJAST literally says "to say something about oneself". Usually, a reflexive directs an action toward oneself, so one might think that this would mean "telling oneself something", which is close but not right. It slightly modifies what one would expect, meaning: "telling (someone) about oneself". It does not make one the object of the message (receiver as well as a sender, as one might expect from a reflexive middle voice), one actually becomes the subject-matter of the message. So, why "reflexive"? One is not telling oneself any thing or talking to oneself? Well, despite being totally the agent of the message, you kind of send yourself along with it. The message is about the sender. "To say something about oneself" is the complete rendition of SEGJAST. It can thus be regarded as a variety upon the reflexive middle voice, or a curiously refined version of it. 

NUANCES: Taking the first verb first I give you examples: 
"Hann segist (vera) stór (= he claims to be big)" 
"Hún sagðist ekki vilja fara með okkur (she said she didn´t want to come along with us)" 
"Ef tveir menn segjast vera Jesús, þá hlýtur annar að hafa rangt fyrir sér (= if two men claim to be Jesus, then one of them must be wrong" 
You can follow up to SEGJAST with an adjective (without necessary using "vera") or a verb in infinitive or a noun (preceded by a verb in infinitive). As with the English verb "claim" certain unreliability is suggested by making the statement so openly second-hand. 

LITERAL ANALYSIS: The other SEGJAST is a passive. What has to be said about it is that it is only used passively in certain fixed impersonal phrases, which should be memorized: 

Það verður að segjast að... 
Það verður að segjast eins og er að... 
 

These two mean the same, i.e. "one has to admit that... (lit: it has to be said that...)". Like you see, the context of the usage is something that is about to be said, as an excuse of sorts. A plain statement using the passive saying that something has been said, i.e. "það sagðist" would definitely sound awkward. We say "eitthvað var sagt" instead.