It may seem counterintuitive that cutting off armies has any effect in Risk -- after all, in real life, cutting off an army mostly hurts that army by depriving it of supplies. But cutting off armies can be very valuable in Risk.

A skilled player uses armies in mutually supporting ways; in particular, multiple adjacent armies must all be beaten by a penetrating opponent who wants to hold the territories he just took. A cut-off army can't easily be used to support other armies. A cut-off army either needs to expend its strength fighting its way back to its own lines, resulting in a smaller army in the critical theater, or sit still. A strong enough army just sitting there may be immune to attack, but, without a massive numerical advantage, a cut-off army won't be able to go on the offensive.

Even better, cutting off your enemy's army presents him with a dilemma: reinforce that army so that it can survive or attack, depriving the critical theater of those resources; or abandon the cut-off armies, precious production that they may represent.