What is the maximum wind speed a Boeing 737-800 can deal with during takeoff and landing?
As your source shows, the maximum tail wind is fixed and fairly simple.

Getting slightly more complex, there is technically no maximum headwind I'm aware of, although most airports will close when the wind gets much above 50 knots, and even with the wind coming almost straight down the runway, you still get some crosswind component, which factors in. Expect to see many pilots divert in winds much above 40 knots even if the airport is open, especially if gusting to above 50.

Crosswinds are a little different as they depend on the braking action - basically how grippy the runway is. Simplified, you can think of it as a scale from dry weather, through slightly and very wet, then slush and ice. The grippier the runway, the more the tyres will grip and the higher crosswind the aircraft can deal with. The higher numbers in the chart you show are for a dry runway (good on your chart) the lower numbers for wet or icy (poor on your chart) runways.

Although it's written in terms of 'braking action', it's really a measure of grippiness, and it's important both when braking, but also for the ability of the tyres to prevent the aircraft sliding laterally off the runway. This job doesn't solely come down to the tyres, but they're part of it. This is also a large part of the reason why landings and takeoff have different maximum crosswinds on some aircraft.