We consider high-order connectivity in $k$-uniform hypergraphs defined as follows: Two $j$-sets are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. We describe the evolution of $j$-connected components in the $k$-uniform binomial random hypergraph $\mathcal{H}^k(n,p)$. In particular, we determine the asymptotic size of the giant component shortly after its emergence and establish the threshold at which the $\mathcal{H}^k(n,p)$ becomes $j$-connected with high probability. We also obtain a hitting time result for the related random hypergraph process $\{\mathcal{H}^k(n,M)\}_M$ -- the hypergraph becomes $j$-connected exactly at the moment when the last isolated $j$-set disappears. This generalises well-known results for graphs and vertex-connectivity in hypergraphs.