TL;DR: Water is incredibly easy to get and work with.
Water is a good solvent for polar compounds, and the reasons for this are laid out pretty well by John Snow, but that's not really what makes it the universal solvent.
Instead, a series of other, incidental properties makes it a popular choice:
There's a lot of water. It literally falls from the sky. This means that for most applications, it's the absolutely cheapest solvent available.
Water is a reasonably stable molecule. Few things you can dissolve in water are expected to react significantly with it. It won't readily burn or combust, even in vapour form, and its high heat capacity and prevalence (see above) means that it can effectively douse most runaway reactions that might occur, such as fires (there are times when adding water to a fire is a bad idea, especially perhaps in a lab, but even those times the problem can be solved by adding a lot of water).
Unlike some other common solvents, water is not toxic, and is not a bio- or eco-hazard. This means that unused solvent can be disposed of without special precautions, and if the solute can be disposed of trivially, the solvent can as well.
Purity and stability
Water is everywhere, giving it another unique property: Unlike many other solvents, water won't absorb atmospheric water vapour, decreasing its purity as a matter of course. (It'll still exchange water with the atmospheric vapour, of course, but it doesn't significantly change its make-up.)
Distilled 100 % water will remain more or less pure unless something is actively added to it, which, while not unique, is useful and somewhat uncommon. Water molecules will also not spontaneously decompose under most sane conditions.
Water is liquid at a wide range of temperatures, stretching from 0°C to 100°C. Under laboratory conditions of ~1 atm and 20-25°C, water is a well-behaved and practical liquid.
Furthermore, while the liquid phase is available at a wide range of temperatures, the solid and gaseous states are not out of reach when using even extremely basic equipment. Cooling water to well below freezing, say -18°C, or to boiling, are both trivial activities, using equipment that is not only available in every lab, but even most homes.
Water isn't always the best tool for the job, but it's almost always a good or great one, at least when working with polar compounds. A common alternative to water is ethyl alcohol, which has similar polar properties and operating temperatures, but ethanol does have weaknesses in the other categories mentioned here, i.e. its reactivity, its absorption of water vapour, and its price.