How did mariners measure boat speed with a rope with knots tied in it?

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You have a line with knots in it and a float on it. You throw it overboard and turn your sand time piece over at the same time. You count the number of knots on the line that go through your hand in a minute.

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the rope with the knots in it, lol, didn't measure speed. it measured the depth of the water and the type of bottom.

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The device that sailors used to make their speed measurement was called the "chip log." Chip as in chip of wood, and log as in to record in a log. The chip was a wedge of wood about 18" in size; it was tied to one end of a rope on a large spool. The rope had knots tied into it about every 47'3" (more about how that was calibrated below).

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The wooden chip was thrown overboard at the ship's stern (back end). Because of its wedge shape, it would "grab" the water and start pulling out rope as the ship moved forward. One man would hold the spool of rope as it played out; another man would start a sandglass filled with 30 seconds of sand; and a third man would count the knots as they passed over the stern board.
When the 30 seconds of sand expired, the time keeper would call out and the counting of knots would stop.

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The faster the ship was sailing, the more knots and the longer the length of rope was played out. The number of knots in the rope that were counted in 30 seconds, then, was equal to the speed of the ship in nautical miles per hour. A "knot", therefore, is not a nautical mile, it is a nautical mile per hour. Thus 1 knot was equivalent to 1 nautical mile per hour; 5 knots were equivalent to 5 nautical miles per hour; etc.

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Finally, the actual values of 47'3" between knots on the rope and the 30 seconds that were used with the chip log break down like this;-

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The length was based on converting 1 nautical mile per hour to feet per second(fps), and then multiplying fps by 30 seconds (which was a practical time to spend counting knots with a sandglass). The result was the calibrated length in feet at which to tie the knots for a 30-second run of the chip log.

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... Now that we have much more precise technology to measure things, and because we need to establish international standards and conversion factors, and because the earth is not uniformly flat or round anywhere, and because even the precise definition of the second has changed, the official value for how many international feet in an international nautical mile has changed.

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Likewise, the measuring time of about "30 seconds" in the sandglass is now calibrated at 28 seconds. Time itself didn't change by 6.7%! But the cumulative effects of new international standard definitions for time, feet, statute miles, and nautical miles and more accurate measurements of the actual size of the earth -- and the fact that we don't use sand to measure time anymore -- have changed the modern calibration of the chip log.