laced Edwardian huckaback linen guest towels (or "show towels"). No defects just light staining on one.
That specific grid-like, honeycomb texture is known as a huckaback weave (often just called "huck linen"). It was intentionally woven with small, raised geometric blocks to make the linen highly absorbent and durable. It is notoriously long-lasting and crisp to the touch.
Lace on Only One Lower Edge: These were historically used as "guest towels" or "show towels". They were hung over a washstand rail or a bathroom towel bar with the decorative lace end folded and draping prominently outward to impress visitors. Leaving the other side plain made it easier to loop over the rack.