Why does my footnote not show up at the bottom of the page?

Inside displayed equations, the usual footnote command doesn't work. Instead, you need to (a) insert a footnotemark directive where you want the footnote marker to be placed and (b) insert a footnotetext{...} directive immediately after the end of the math material.

That said, and echoing @pluton's earlier remark, I don't think it's good practice to "bury" such information in a footnote. Instead, I think it's better to write an extra explanatory sentence or two in the body of the document right after the equation. Something like:

The final equality in the preceding sequence of equations follows because we assume a balanced budget, therefore $mathit{Tr}_t=0$, and because we further assume that income taxes and bequest taxes have the same effects in long run.

documentclass[12pt]{article}
usepackage[T1]{fontenc}
usepackage{mathrsfs}
usepackage{amsmath}
setlengthtextheight{7cm} % just for this example

begin{document}
begin{equation*}
begin{split}
    bar{c}_t 
    &= int_j c_{t_j},dj 
    &= int_jbigl[R(1-tau_{B_{t+1}})b_{{t+1}_j}
       +(1-tau_{Y_t})w_{t_j}h_{t_j}+mathit{Tr}_t-b_{{t+1}_j}
       -h_{{t+1}_j}bigr],dj 
    &= R(1-tau_B)b_t+(1-tau_Y)y_t+mathit{Tr}_t-b_{t+1}-h_{t+1}  
    footnotemark  
    &= Rb_t+y_t-b_{t+1}-h_{t+1},.
end{split}
end{equation*}footnotetext{Here we assume balanced budget, 
therefore $mathit{Tr}_t=0$. We also assume income taxes and 
bequest taxes are same in long run.}

Bla bla bla dots
end{document}