You got it in one – fuel management.
To fly a straight path would require large amounts of fuel to reach the appropriate speeds. The problem is that the more fuel your carry, the larger and heavier your rocket needs to be, and the harder it is to launch. So instead we look to find every way possible to reduce fuel expenditure.
One of these is to utilise the gravity of things like moons and planets. The gravity will attract the craft and boost its speed as it gets closer, and then we can use relatively short bursts of the engines to escape from that gravity will, now travelling at a much faster speed (but having spent much less fuel).
For longer journeys like the deep space probes, it can even be possible to plan a route to do this multiple times with different bodies in a row.
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