From Sonic Retro
The Egg Walker is Tails' final boss in Sonic Adventure. After plucking up the courage to face Robotnik on his own, chasing the Egg-O-Matic through Speed Highway to disarm the fat man's missile, Tails ambles out into the casino district... to be faced by this towering behemoth of mechanical weaponry. The fox's self-assurances that he's not afraid fool no-one.
Like most bosses in Sonic Adventure, the Egg Walker changes its attack patterns depending on how much damage has accumulated in its 5-hit health bar. Over the course of the fight, Eggman launches exploding bombs, blazing flamethrowers, and sends out shockwaves from his stomping feet after spider-crawling back and forth across the fight arena. To defeat him, Tails has to strike at the glowing blue ankles of the machine just after one of these stomp-shockwaves has been fired. A Tail Swipe is usually a good way to go about it. This doesn't damage the Egg Walker directly, but causes Eggman's cockpit to slump forwards so Tails can land a spin jump.
With Robotnik eventually defeated, Tails receives cheering adulation from the gathered crowds of Station Square. Saving the city from Eggman's missile, and then defeating the scientist himself in this battle, resurfaces as a plot point in Sonic Adventure 2, when Rouge shows Eggman a newspaper indicating that Station Square later rewarded Tails with a Chaos Emerald. The fat man promptly sends Rouge to go steal it.
In all versions of the game, including the DX remakes, if the one presses the whistle/flute button, one can manipulate the Walker's actions. It's unknown at this point what exactly the button does, but it seems that it forces the boss to perform the next action in the cycle. For instance, if the Walker is stopped and firing on an edge of the arena, and the player is on the other edge, pressing the button will make it start walking to the other side. This might be a leftover debug feature, reminiscent of the Y-button board function in Metal Harbor.