Agreed that this is pretty terse. The sequences they're talking about are called Cauchy sequences [0]. A sequence a_i is Cauchy if for any epsilon, there exists an N such that if m and n are both greater than N, then |a_m - a_n| < epsilon. A classic example, suppose your space is the set of rational numbers, and consider the sequence a0 = 1, a_1 = 1.4, a_2 = 1.41, ... a_n = sqrt(2) up to n digits after the decimal place. You can verify that this is a Cauchy sequence, successive points get arbitrarily close to each other. This means the rational numbers are incomplete, because this Cauchy sequence of rationals doesn't converge to a rational number. It's the real numbers that forms a complete space.

Completeness is required for nice results like the spectral theorem for self-adjoint operators [1] to hold, which is pretty essential for Quantum Mechanics.

[0] https://en.wikipedia.org/wiki/Cauchy_sequence.

[1] https://en.wikipedia.org/wiki/Spectral_theorem