To understand why this is so, see
This is a long book [685 pages] and it is difficult to summarize a work of this length and depth, but IIUC his basic findings are:
(1) The golden thirty year period from 1950-1979 when many of us grew up, was not normal but exceptional, primarily because of the destruction or consumption of 50% or more of existing capital in WW1 and WW2, and the boom caused by the reconstruction of war damage. Because of the low [relatively] amount of capital, a much larger than "normal" share of national profit went to labor.
(2) After 1970, in large part due to Reagen and Thatcher, national private capital stocks [particularly large hereditary blocs, e. g. Koch Brothers] began to grow, and due to compounding, rapidly attained their pre WW1 levels, which meant that increasing amounts of national income were going to capital and not to labor.
(3) Because of globalization, supranational corporatism rapidly grew, promoting widespread tax avoidance/evasion again increasing the amount of capital accumulation and thus capital's share of the national income. Additionally, the political influence that large amounts of wealth provides was used to promote "trickle down" and "supply side" economics and the Laffer curve, resulting in even faster growth in high net worth individual capital, due to increased income and compounding.
(4) Unless the rate of capital accumulation by the "lucky few" can be limited, and indeed reversed, there will be continually less available for labor's share of the national income. The late stages will be a oligarchy/plutocracy rather than a representative democracy, which will collapse, when capital's share of the national income is so high, that labor's share no longer provides subsistence. This problem has been compounded by several states allowing perpetual trusts, which have been illegal since the middle ages.
(5) For an educator, the importance of Piketty's work is that education and training are no longer a solution, or even a palliative, for the large majority, unless the rate and amount/fraction of capital accumulation can be limited and as possible reduced.
FWIW -- As a conservative, I find Piketty's solutions of progressive levies on all types of capital above a few tens of million dollars, much steeper progressive income taxes, elimination [or at least severe restriction] of the capital gains tax preference, and international tracking of assets and capital flows to be about as appealing as a double root canal, but the replacement of the Republic as we knew it, with an oligarchy is even less appealing, and least appealing of all is a bloody revolt as occurred when the French endgame played out in 1790, and the rentiers lost not only their capital but also their heads.