Why was it fitting for Jesus to be baptised to "fulfill all righteousness"?

I wondered the same thing, and when I looked quickly at an online Strong's Concordance, the meanings of one of the original Greek words for righteousness is "ordinance" or "decision" ... so maybe it was to fulfill something ordained by the Father.

I've always wondered about that too!

Fulfil makes me think first of fulfilling the OT prophecies, which we know is what Jesus did, and what Matthew tried to bring out lots in his writing. So maybe Jesus was supposed to get baptised, even perhaps by the one whose voice was making ready his way (ie. John the baptist)? I dunno...are there any OT prophecies about that?

From my understanding, it was necessary for Jesus to be baptized so that He would receive the Holy Spirit - after that we see the Spirit leading Jesus around. Jesus needed the Spirit for His ministry and to identify Him as the suffering servant, so presumably without the Spirit, Jesus wouldn't have been able to do much fulfilling of anything.
Feel free to correct me here though.

Ming, Matthew is quick to point out an fulfillments to the OT, but in this case he doesn't and uses the curious statement "all righteousness". I'm not sure what to make of it, for John's baptism was a baptism of repentence (and Jesus had nothing to repent of), and Jesus was of course already all righteousness.

Ralph, no I'm sure he already had the spirit. I can't remember any verses now, but if John the Baptist had the HS from the womb, surely Jesus would too?

Yes, John had the Spirit since the womb, but that doesn't mean that Jesus did. In all the synoptic gospels, the Spirit comes/descends upon Jesus, and immediately starts leading Him. Particularly in Luke (see 3:21,22 and 4:1) there is the description on Jesus coming back from the Jordan "full of the Holy Spirit". Then in John's gospel, John the baptist testifies that the Spirit descended and remained on Jesus and if Jesus already had the Spirit, why does He need to come down and remain on Jesus?
This sort of reminds me of Judges, where Samson didn't have the Spirit until it descended upon him.