。
As the weather got colder, his condition kept getting worse.
Literal
Weather [subject-が] cold-become [as-にれて] he [possessive-の] condition [subject-が] more-and-more was-bad.
~につれて ('as / in proportion to') is a connective marking correlated change — 'as X happens, Y happens accordingly.' 寒くなる ('get cold') is the い-adjective 寒い + くなる ('become'). ますます ('more and more / increasingly') emphasizes the progressive worsening. 具合 ('condition / state of health') is the everyday word for physical wellbeing.