Being a mod for the first Doom, the Bag of Spilling primarily comes from the episodic nature of the plot. Yet, it actually makes sense in light of how the post-boss narration describes what happens during the transition:
You don't carry much at the start of "The Shattered City" because you just started your dream journey.
Before starting "The Stagnant Fortress", the Traveler takes a ship to reach the next place of his journey. He probably wasn't allowed to carry all his stuff onboard.
After vanquishing "The Stagnant Fortress"'s boss, the Traveler has a hallucinatory episode which leaves him delirious for week. He resumes his journey after being nursed by friendly ghouls; they brought him back to their place but obviously left his arsenal at the place he fainted. That's why you reach "The Land That Time Forgot" unarmed, again.
After beating "The Land That Time Forgot", you're brought to "The Plateau of Leng" by Nyarlathotep's teleportation. Since he's a malicious being, he obviously made sure you'd lose your arsenal on the way.
And last, you start "Out of the Aeons" without your previous gear, again, because it's another dream journey which starts several years after the Traveler eventually managed to leave the Dreamlands.
The mod's version of the Dreamlands features more modern/futuristic elements than described in Lovecraft's canon (for instance, most of the second episode is set in a derelicted metal fortress). But the point of the Dreamlands is that its content depends on the dreamer themselves. The time era the Traveler comes from isn't specified, but the mere fact he encounters Kalashnikov-toting enemies would imply he's from the second half of the 20th century (if not even later), justifying the more modern things encountered, either borrowed from things he encounters in his mundane life or from things he knows from Speculative Fiction.
