Q:

Why do I need Bridge mode on my main router to connect a second one? 

I want to buy a better router than the one my ISP provides me.  I can’t just ditch my ISP router because it is modem and router at the same time. 
What I’ve been told is that I would need to set my ISP router in bridge mode to have a second one managing all my home network. 
My ISP router doesn’t have a bridge mode option.  I’ve been told that I could use a DMZ configuration and it would be the same. 
My true question is why do I need my first router with bridge mode in the first place? 
Example: I connected a old router I have to the ISP router.  LAN port on my ISP router, WAN port on the “old” one.  Addresses of 192.168.1.x in my ISP router, 192.168.0.x on the “old” one.  Everything seemed to work fine on both routers.  I was told that with this configuration I would have no internet access in the “old” one, but I had. 

A:

Bridge mode from a modem/router is a cleaner, less risky setup than router-to-router. 

“My true question is why do I need my first router with bridge mode in the first place?” 

You answer your own question at the beginning when you say:

“I can’t just ditch my ISP router because it is modem and router at the same time.” 

So you don’t want the router functionality.  But to bridge between whatever connection your ISP has to your new router, it is best practice to just (effectively) disable the router aspect of your ISP provided modem/router so the device is just a modem. 
As a modem, it bridges whatever their connection/protocol from the ISP to you is — coaxial cable with DOCSIS or RJ45 with ADSL and such… — to Ethernet protocol your router can handle. 
Again, this is the best practice since there is one step from ISP to Ethernet and then from Ethernet to your new router.  But if there is no obvious Bridge mode on that modem/router your options are limited as you explain. 

“I’ve been told that I could use a DMZ configuration and it would be the same.” 

Well, it would technically “work” but you introduce a layer of complexity that is really unnecessary at best, problematic at worst. 
Connecting that modem/router to another router means you are hopping between two networks int your home to get outside.  If done right, it can work fine without any issues.  But if not, you could end up with — for example — two DHCP servers fighting it out and confusing the network.  Which could lead to slowed down connections at best, crazy and unpredictable “outages” as the two routers fight to control the network at worst. 
If you can’t really get a true Bridge mode on that modem/router then I would recommend the following steps to neuter network management on the modem/router:

Disable DHCP on the Modem/Router: Your new/preferred router should handle that. 
Assign a Static IP Address to the New Router: With DHCP dibbled, assign a static IP address to from the modem/router to the new/preferred router. 

That should work. 
But again, it’s just there is always a risk that your ISP provided modem/router might get in the way of things.  So it’s always better to just go to Bridge mode. 
Inquire with the ISP if there is anyway for you to exchange the modem/router for a plain modem.  I doubt that option exists with your ISP based on what you have described, but still it is worth asking about.