Master Windu was a mighty Jedi Master (even Count Dooku couldn't stand a chance against him), and this fact should be popular among the bad guys, if not across the Galaxy. What made Jango Fett think he could take out Master Windu all by himself? Yes, he was flying, and Windu was on the ground, but this hardly matters in a universe with The Force.
Fett was hardly alone when he jumped into the arena -- a huge battle was in progress in which all the Jedi were fighting an army of battle droids as well as the animals intended to kill Anakin, Obi-Wan, and Padme.
Windu had just lost his lightsaber to a reek the moment Fett jumped into the arena, and Fett jumped in to try to grab Windu's lightsaber. Fett was trying to take advantage of Windu's momentary distraction by the reek and the loss of his lightsaber. Presumably, Fett would have used his jetpack to retreat back to a safe distance, and Windu wouldn't have his lightsaber to deflect Fett's shots (or anyone else's).
Unfortunately for Jango (and poor Boba), Windu was able to use the Force to grab his lightsaber back just before Fett reached it, and then Fett was immediately trampled by the reek and his jetpack destroyed. Fett can't use the Force, so he couldn't know ahead of time that Windu would beat him to the lightsaber by just a moment and that the reek would turn on him and destroy his jetpack.
Fett fought well against Obi-Wan on Kamino and had defeated other Jedi, so he probably felt he would be able to defeat Windu with the help of the battle droids, animals, and the fact that Windu wouldn't have his lightsaber.
The novelization addresses that issue. Please note that the most important factor is #3 combined with #1:
Mace "started it" by holding a lightsaber to Fett's throat (meaning, Fett knew he should attack Mace while he was busy with droids - and temporarily unarmed, as shown in the next bullet point), and not wait till he was done and attack Fett one on one.
The Count's expression changed, though, just for a moment, as a familiar hum began right behind him. He glanced to his right quickly, to see a purple lightsaber blade right beside Jango Fett's neck, then turned slowly to regard the wielder.
As such, Fett already started fighting Mace...
Mace Windu spun about, recognizing that his most dangerous enemies were behind him. He faced Jango Fett--and found himself looking down the barrel of a stout flamethrower. A burst of flames reached out for the Jedi Master, igniting his flowing robes.
Fett went to the arena to fight Mace Windu - as indicated above - because he needed to do so before Mace was free to attack him again - AND because Mace was NOT armed at that moment - disarmed by the reek:
The reek followed Mace, and he had to slash wildly to fend it off. He did manage to drive it back, but was butted and lost his lightsaber in the process. He came up facing the reek, and figured that he could outmaneuver it to get his weapon back easily enough, but then an armored rocket-man flew down in his path, blaster leveled.
... Mace reached out with the Force and brought his lightsaber flying to his hand, moving like lightning to parry Jango's first shot.
It clearly shows that Fett was NOT trying to engage Mace short distance, but was trying to widen the gap, the moment Mace recovered his lightsaber:
The Jedi was on him immediately, lightsaber weaving through the air. Jango dodged and lifted into the air with his rockets, trying to keep one step ahead of that deadly blade and to occasionally fire a bolt at Mace.
I don't think there's a canonical answer (although the novelization might offer some hints as to his thoughts), but it's probably because he's a Mandalorian and Mace Windu is a Jedi. Due to the Mandalorian Wars, there's a lot of bad blood between the two groups. Combine that with a fair degree of arrogance in his combat prowess, and he leapt into the fray.
Quoting from the Wikipedia article, it seems that Jango had a particular reason to hate the Jedi.
Jango took command of the remaining True Mandalorians, but unknowingly led them into a trap at the Battle of Galidraan, where they were framed for murder, resulting in the complete destruction of the True Mandalorians at the hands of the Jedi. Jango is the only survivor, and before being captured, slays half of the members of the Jedi Council as well as a few Jedi Knights single-handedly.
The original film script (as written) indicates that the fight between Fett and Windu was pretty much continuous. My guess would be that it's only down to an editing decision that it looks like there was a delay in his jumping into the arena after Windu.
COUNT DOOKU signals. THOUSANDS OF DROIDS start to pour into all parts of the arena.
MACE WINDU draws his lightsaber, JANGO FETT draws his guns and fires at MACE WINDU, who deflects the shots. JANGO FETT and MACE WINDU jump into the arena, where they fight. BOBA FETT watches his Dad and the Jedi Master fight. The battle begins.
GEONOSIANS fly away everywhere. DROIDS fire at JEDI, who deflect the bolts and cult down the DROIDS. The GEONOSIAN TROOPS fire ray guns that are more difficult for the JEDI to deflect.