There are many benefits to a properly implemented warmup routine:

Improved nervous system function
Increased sensitivity in nerve receptors

Faster nerve impulse transmissions

Increased core and muscle temperatures
Lower level of muscle viscosity

Improved blood flow to muscles

More efficient uptake of oxygen from blood

Improves muscle flexibility and joint ROM

More forceful contraction and rapid relaxation of muscles

Other benefits
A more thorough lubrication of joint structures

Acclimation of soft tissue to increased loading

Activation of inhibited muscle groups

A marked decrease in the chance of sustaining an injury

Sample Warmup:

No informative writeup would be complete without an example. The sample person here has very weak and inactive gluteals, has internally rotated humeri, and is very tight in the hamstrings, hip flexors, calves, chest, and lats:

Foam Roller:
Hip flexors
IT band
Calves
Lats

Activation Circuit:
Supine Glute Bridges “ 2x10 - 3sec isometric at top (Glutes, duh)
Band Resisted Sidesteps “ 2x12 Each Direction (Glute medius/minimus)
Shoulder Dislocations “ 2x12 (External rotators)
Wall Retractions “ 2x12 (Scapular retractors)

Dynamic Flexibility:
Inch Worms - 2x6
Dorsiflexed Ankle Mobility - 2x12
Shoulder Circles - 2x12 (Each direction)

(Note: Someone requiring this much remedial activation work probably cannot move this kind of weight, but just humor me for example sake)

Specific on Leg Day:
Deadlift 1RM Attempt
95 x 5 No rest
135 x 3, No rest
185 x 3, No rest
225 x 1, 30s rest
255 x 1, 60s rest
285 x 1, 60s rest
305 x 1, Full Recovery

Back Squats
135 x 5, No Rest
185 x 3, 60s Rest
Worksets:
205 “ 3x10

Romanian Deadlifts
135 x 5, No Rest
185 x 3, 60s Rest
Worksets:
225 “ 3x8

DB Lunges
5s x 3, 60s Rest
Worksets:
30s - 3x12