How do we know that the universe is currently expanding, who proved it and how?

By measuring the redshift in the spectrum of special marker stars called Cepheids, which are reliable emitting light in the same spectrum and which have a known relation between variable period (they don't emit light constantly, but the intensity varies over time) and luminosity. Polaris, the northern star, for example, is the most famous Cepheid.

The further away a galaxy was, the more the spectrum of the Cepheids was red-shifted (the doppler effect made the wavelengths longer). By using the also known light emission (luminosity) of the Cepheids for calculating the distance, Edwin Hubble was able to create a correlation between the speed at which a galaxy is moving away from us and the distance to us, called now Hubble's law. Each Megaparsec (about 3.26 million light-years) distance between us and the other galaxy expands by 70 km every second - very slowly, but noticeable over longer distances.

That is also why it's effect on galaxies is not too big, they can easily move faster through the expanding space, than space expands and thus collide with each other or orbit around each other in clusters.

80 years ago in1929, Edwin Hubble (the astronomer who the space telescope is named after) observed and subsequently proposed this theory since the spectra of stars in observed galaxies was shifted toward the red end of the spectrum.

The so called 'red shift' is the light wave version of the sound wave shift discovered by Christian Doppler, called the Doppler Effect.  This effect is explained in sound that a moving source is observed to have a different frequency than what is observed when it is stationary.  
   If the source is moving toward the observer, the incoming wave front is 'compressed' resulting in a higher pitch heard.  When the source is moving away from the observer, the waves are 'stretched out' resulting in a lower observed pitch.  

   Note that the source is not really changing the pitch, it is the relative speeds (toward or away) that produces the effect.  This can be observed by the change in pitch you hear from a passing emergency vehicle's siren.  It is not just the intensity of sound that changes, but the pitch.

   Hubble reasoned that this same effect would apply to light waves.  The observed spectra of stars in galaxies did not match the light one would expect. When the spectra was slid over toward the lower frequencies, everything matched up.  Red light has the lowest frequency in the visible spectrum, so Hubble reasoned that the observation of a 'red shift' meant that the objects themselves must be moving away from the observer.  Subsequent observations led astronomers to believe that most everything was moving away from everything else!

This theory implied that the universe was not static and if one 'ran the
program in reverse' it would lead to an inevitable 'beginning point' which meant there was a Genesis Event to begin all things.  The discovery in the 1960's of the 3 K 'hiss' supported this concept of a grand explosion In The Beginning that results in what we observe today.

   Even more mysteriously, recent observations suggest an ACCELERATION of the movement of objects away from one another, which has given rise to the theory that dark matter may be responsible for the push, implying an antigravitational force, which may not be so surprising as other fundamental forces (electromagnetic, for example) exhibit positive and negative values.  

   The fate of the universe, then is also in the balance, for if there is a constant force being exerted then distances between objects will continue to increase without cease.  If there reaches an equilibrium point, and 'positive' gravity prevails, then a 'blue shift' is theorized ultimately resulting in a big crunch which may ignite another explosion.
This is an 'oscillatiing' universe theory, meaning the universe periodically undergoes expansions and contractions with unknown intervals between events.  The best we can say now from current observations is that the distances are increasing, and only G-d knows
the ultimate fate of the cosmos.