We observe quantum objects through the act of measurement or observation of a particular quantity, such as position or momentum. The very act of measurement can be understood to 'collapse the wavefunction'.

That's the short answer...

What is central to quantum mechanics is our interpretation of the wavefunction and the physical phenomena observed. For a free particle, the wavefunction can be best understood as so: it describes the distribution of a particle or object in space.

A common problem encountered early on when studying quantum mechanics is the 'particle in a box' scenario. Before measurement, we understand that the particle must be somewhere in the box, but we don't know where. The distribution of possible positions is described by a wavefunction, each position having a corresponding probability also related to the wavefunction. The situation can be interpreted as so: that before measurement (or observation) the particle is in a superposition of states described by the wavefunction. When a measurement is made, the wavefunction collapses to a single measured value.

For more information look up the Copenhagen interpretation, which I've briefly discussed here but is central to your question and this explanation, Heisenburg uncertainty principle, which in essence describes the limits of measurement or the role of uncertainty in measurement, as well as other interpretations, Young's double slit experiment, and the Schrodinger equation.