There are two essential factors that make a space telescope a good idea.
First of all, it gets above the atmosphere. This has the effect of removing three major obstacles to ground-based observation.
1: The weather. No cloud cover or rain can come over and ruin your observing session. The sky is always clear.
2: The sky itself. A telescope is useless during the day, as the
rayleight scattering of sunlight that causes the sky to be blue blots
out the stars entirely for many hours each day. The Hubble telescope is
not restricted by day or night cycles.
3: The air causes distortion and so severely limits the resolution of
any image obtained by looking through it. Getting above the air removes
this distortion, allowing sharper images.
The second factor: it gets it off the ground. No matter where you put
your telescope on the ground, it can only point at a given bit of sky as
long as the Earth is turned towards it. If you put it in space it can
aim at any point in the sky for hours or even days on end. Because the
distances of the objects it is used to study are so vast, the fact that
it is constantly moving around the Earth in its orbit makes no
difference to the apparent positions of most things and hence it can sit
and collect light from the same spot for a very long time, allowing
incredibly faint and distant objects to be imaged.
An image like the Hubble deep field could never have been taken by a
ground-based telescope. It took days of continuous exposure to collect
enough light to make those distant galaxies visible. The Earth's
rotation, the daylight, the weather, the atmospheric distortion would
all have ruined it
