Because they think Savage is the only hope for the world
Kind of funny, actually:
Druce: In 2175, less than ten years from when Savage conquers the world, the Earth is attacked by a warlike race of extra-terrestrials from the planet Thanagar. Without Savage to unite the world under a singular rule, [fire and destruction appears on a screen] this is what becomes of the world.
Druce: This is what happens to the Earth is Savage is not there to lead it. Without him, all human life is extinguished. Even [the Vanishing Point] falls to the Thanagarians.
Legends of Tomorrow Season 1 Episode 15: "Destiny"
Most immediately, because Rip and the Legends are breaking The Rules, and there are consequences for that:
Time Master Druce: These individuals have manipulated the timeline without authorization on multiple occasions. You [the Pilgrim] are hereby ordered to hunt down these individuals' younger selves and eliminate them.
Legends of Tomorrow Season 1 Episode 12: "Last Refuge"
Rip, for his part, knows he's breaking the rules; he just thinks that the benefits of his actions justify his transgressions.
What's the deal with those "rules", anyway?
Let's get some more context on that first quote I gave:
Time Master Councillor: As a Time Master of long standing, you seem to forget our responsibility is to protect the timeline, not humanity. You wish to engage in the very sort of manipulation of time that we are sworn to prevent.
Rip: And yet, if we allow humanity to perish at the hands of evil, then what universe are we custodians of?
Councillor: What if, in trying to save the world from Savage, you make it vulnerable to another, more terrible despot?
Legends of Tomorrow Season 1 Episode 1: "Pilot, Part 1"
It all comes down to philosophy, basically.
The Time Masters believe in the burden of power: they can change anything they want in history, but (to paraphrase a great philosopher), just because you can do something does not mean that you should. One of their main concerns is that changing the timeline can have devastating, hard-to-predict consequences1; we saw this in the episode "Star City 2046", where it's suggested that Ray and Sara joining Rip's crew contributed to Star City descending into an anarchist dystopia.
Rip Hunter, on the other hand, believes in the responsibility of power. His thesis is that, because the Time Masters have the ability to improve the lives of human beings, they have a moral obligation to do so.
The Time Masters are so determined to stop Rip and his crew because of the effect their actions are having on the timeline:
Time Master Druce: End this foolish crusade return with me to the Vanishing Point. You'll be acquitted of all charges, and in a few hundred years, we may even be able to fix the damage you've caused.
Legends of Tomorrow Season 1 Episode 4: "White Knights"
1 This is something I've talked about on this site before, though in a different fictional universe; the main points of my thesis over there still apply here, however