I've always wondered, that since the Earth is moving at a very fast velocity around the Sun, why is it that when astronauts leave the Earth, the Earth doesn't immediately move away from them at extremely large speeds?
Because you were also in orbit around the sun with the Earth and still have that velocity.
You may be imagining this in terms of stepping off of a slow moving vehicle on the Earth: you jump off, you come to a stop relative the ground and watch the trolley car go it's merry way. But that is a feature of friction between you and the ground. There is no such thing as a absolute reference frame in the universe and when you "leave the Earth" you don't come to stop relative anything so that you can watch the Earth fly away.
Newton's laws apply here: "a body in motion (that's the you or the planet) will continue in motion unless acted on by an external force". You just keep going except for changed induced by your drive.
Because until you're far enough to leave Earth's gravitational field, you will still feel it. That means that if you go into space (say at ISS level) and put on a spacesuit and go for a walk, you'll still be moving with the Earth as it rotates around the Sun.
When you've traveled far enough that the force of Earth's gravitational field is negligible, you will see movement.
The title question and the text question are different. The title question "Why doesn't the Earth leave you as soon as you escape it?" has the answer: the Earth does leave you as soon as you escape it. The definition of "escape" is greater than escape velocity (around 11 km/s), which means that you no longer orbit the Earth. You and the Earth will be moving away from each other.
Unless you have also reached escape velocity from the Sun (about 17 km/s from the Earth), then you will still be orbiting the Sun, and so with no other maneuvers you will come close to intersecting Earth's orbit every one of your orbits (depending on the effect of other planets, mainly Jupiter), and you may eventually reencounter the Earth depending on the phases of your orbits.
The text question asks about astronauts leaving the Earth. To date, astronauts have never reached escape velocity, and the vast majority of them (all but 24 of them) have been confined to low-Earth orbit (about 8 km/s). So the answer to the text question is that astronauts (so far) are bound to the Earth by their orbits.
Imagine that you're sitting in a car that's sitting on a wide flatbed truck. The truck is moving down the freeway at, say, 65 miles per hour. You step out of the car onto the bed of the truck. You've just left an object that's moving at 65 mph, but it doesn't move away from you -- because you, the car, and the truck are all still moving at 65 mph relative to the ground.
Now step off the truck onto the freeway. (Don't actually try this!) Your ground-relative speed will rapidly diminish from 65 mph to 0, and you'll see the truck, with the car on it, continue moving off into the distance. (You'll also have numerous broken bones.) This happens because a force was applied to you when you hit the ground. From the point of view of the truck driver, the ground hit you and rapidly propelled you backwards.
When you "step off" the Earth by launching into space, there is no "ground" to hit you and push you backwards. Your initial momentum continues to carry you along in Solar orbit.
We can view the situation from any of several points of view, or "frames of reference".
From a Sun-centered frame of reference (where the Sun is treated as stationary), you start out travelling in Solar orbit along with the Earth, then you leave Earth with enough velocity to escape the planet, but not enough to leave Solar orbit. (Earth's escape velocity is about 11 kilometers per second; Earth's orbital velocity around the sun is about 30 kps.)
From an Earth-centered frame of reference (where the Earth doesn't move but it does rotate), you start out on the Earth's surface, moving at a few hundred kilometers per hour because of the Earth's rotation, then you accelerate and leave the surface. You have, let's say, barely enough velocity to leave the Earth, but not enough leave it at a very high speed. Since we're treating the Earth as stationary, its orbital movement around the Sun can be ignored.
Both of these frames of reference are valid, and calculations using either of them will yield the same description of what happens physically. This is the basis for the theory of relativity, though everything I've described has been known since Newton.