c. 1300, streinen, "tie, bind, fasten, gird;" early 14c., "confine, restrain" (a body part, animal, etc.), senses now obsolete, from present-participle stem of Old French estreindre "bind tightly, clasp, squeeze," from Latin stringere (2) "draw tight, bind tight, compress, press together."
This is reconstructed to be from PIE root *streig- "to stroke, rub, press" (source also of Lithuanian strėgti "congeal, freeze, become stiff;" Greek strangein "twist;" Old High German strician "mends nets;" Old English streccian "to stretch;" German stramm, Dutch stram "stiff").
Strain is attested from late 14c. as "tighten; stretch, extend; make taut; stretch to the utmost tension," also, intransitive, "exert oneself, strive; exert a compelling force;" also "overexert (a body part), injure by overstretching."
The sense of "press through a filter, put (a liquid) through a strainer" to purify from extraneous matter is from late 14c. (implied early 14c. in strainer); that of "to stress beyond measure, carry too far, make a forced interpretation of" is from mid-15c. Related: Strained; straining.
Transitive strain at "make a difficulty of" (1580s) echoes Matthew xxiii.24 (strain at a gnat; Tyndale has the line as Ye blind guides, which strain out a gnat, and swallow a camel), in which the sense seems to be "will strain the liquor if they find (but) a gnat in it."