Actually these are a sub group of the full Lorentzian group. The full group is generated by these transformations and the familiar 3D Euclidian congruence transformations.

That our transformations compose as claimed follows from addition rules for hyperbolic trig functions:
sinh(r+r') = sinh(r)cosh(r') + cosh(r)sinh(r')
cosh(r+r') = cosh(r)cosh(r') + sinh(r)sinh(r')

which can be derived from definitions:
cosh(r) = (er + e–r)/2
sinh(r) = (er – e–r)/2

Given:
x' = x•cosh(r) + t•sinh(r)
t' = x•sinh(r) + t•cosh(r).

x'' = x'•cosh(r') + t'•sinh(r')
t'' = x'•sinh(r') + t'•cosh(r').

We compute:
x'' = (x•cosh(r) + t•sinh(r))•cosh(r') + (x•sinh(r) + t•cosh(r))•sinh(r')
t'' = (x•cosh(r) + t•sinh(r))•sinh(r') + (x•sinh(r) + t•cosh(r))•cosh(r')

x'' = x•(cosh(r)cosh(r') + sinh(r)sinh(r')) + t•(sinh(r)cosh(r') + cosh(r)sinh(r'))
t'' = x•(cosh(r)sinh(r') + sinh(r)cosh(r')) + t•(sinh(r)sinh(r') + sinh(r)sinh(r'))

x'' = x•cosh(r+r') + t•sinh(r+r')
t'' = x•sinh(r+r') + t•cosh(r+r')

Q.E.D.