First, rockets only go straight up at the very beginning of their flight, in order to get out of the atmosphere. After a few minutes they’re going nearly horizontal.
There are two reasons why airplanes can’t go higher. The first is that most planes have a thrust to weight ratio under 1, so they rely on their wings to provide lift. In space, there’s not enough air to provide lift.
The second, and more important reason is that aircraft engines work by accelerating air. They take air in the front, burn it, and shove it faster out the back. If there’s no air coming in the front, they can’t burn their fuel and have nothing to shove out the back.
Rockets on the other hand start with a thrust to weight ratio typically a little over 1, which rises as it burns fuel, and the ship gets lighter(until staging anyway). Once you’re going sideways fast enough the earth curves away from you as you fall, decreasing the required thrust.
More importantly, rockets carry their oxidizer, so they don’t need to bring in air to burn their fuel, and they accelerate by shoving the combustion products out the back. The downside is that you have to carry all that reaction mass with you, you can’t pick it up as you fly like a plane can.
The hard part of getting to orbit isn’t going high enough, it’s going fast enough to not fall back down. The fastest air breathing planes go about 1200 m/s, while you need to go about 8000m/s sideways in order to not fall back down.
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