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In past it came to be known that light behaves both as a wave and as a stream of particles,depending upon the situtation...so then percieving to the symmetry of nature,Louis de Broglie suggested that 'particles' should also behave as a wave..and so according to his hypothesis it is not only electrons,,its me,you, a moving train etc, evry particle behave as wave..
Even Broglie proved Bohr's 2nd law using his hypothesis,which was previously experimentaly obtained...
and so it is that an electron behaves as a wave...and we can even measure it...also it is interesting that we too have a wavelenght but our mass is so huge that it is quite neglegible..and so we behave a much as particles only..hopefully it is explained..
I think we cannot explain why it acts like a wave,it just acts like one, physics can only predict things,it cannot answer why it act like this,by the way ,I think we say an electron acts like a wave because if we use the "wave" model we can explain how it behave ,therefore we say it is a wave
wel in physics there are assumptions and models,,,whicheva suits the best,,is taken as the probable behaviour..and so i think we can explain an electron as a wave...because it suits and explain many observations which cannot be explained on its particle nature,,,which itself is another retriceumption or explanation..
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In the studies it is known that the light has both partical nature and wave nature... similarly debrogli has connected the partical nature of an electron to a wave nature... and said that micro particals have wave nature...