How to explain how Titanfall characters enter the Mass Effect Universe

Use the scene where BT and Cooper plan to sacrifice themselves by detonating BT's nuclear reactor in his chassis causing the ark and by extension the planet to blow up which leads to your character to try to evacuate the ark's/planet's upcoming destruction, while using a crow dropship to escape they suffer a jump drive error as they activate it while BT-7274 simultaneously blows up, therefore, causing the freak accident that sends them to the ME Universe as the ark causes issues with their FTL drive due to the effects of temporal displacement.

A Jump Drive allows a ship to travel very long distances in the blink of an eye. Between jumps, a ship is vulnerable to attack as it plods along using basic thrusters while the Jump Drive recharges. Recharging can take anywhere from a few seconds to several days, depending on the size of the ship and the distance of the jump. The drives require a specially refined fuel in order to recharge, possibly negative matter, and on long series of jumps, there is a real danger of being stranded without fuel, if the ship does not end up within range of a refueling ship or station.

Theory - The Ark and its rings were built by a long-dead alien race who used their tech to easily colonize and terraform the frontier as it is described as a dense zone of closely clustered habitable planets rich with natural resources.

In Effect and Cause, we see that in the present day, the IMC research facility looked like it aged 50 years after the explosion which might support the theory that the ark is meant to terraform worlds by manipulating time to speed up the process.

IMC troops note the interference on Jack Cooper's HUD as he scans the ark and time travels, which is seen in Effect and Cause, which matches the description of the interference caused in the War Games simulations by the alien ruins in the Swamplands. It is possible the Fleet Operations Base is on Typhon.

Use this interference to describe the transition into the new universe

In Revelation Space, they have devices that can mess with Inertia itself, allowing ships to accelerate harder. But if you mess with Space too much, i.e. inducing Tachyonic matter and such, you can get retroactively deleted from the timeline. And nobody will know. You just... disappear, you are cut seamlessly from the timeline. Only people near the device ever realize that someone existed before being erased. And that "Corrective Deletion" goes as far as whole species. Just wham, and they never existed.

To put it more simply, you can think of the relation between time and space as a balance-board. The Universe does not like being toyed around with, and it will make that very clear if you ignore the warning signs. This means that, one can only manipulate time a certain amount in a given area before space responds by acting anomalously. Vice-versa, manipulating space too much will result in anomalous temporal activity. This is because the Ark is a giant fuck you to the world of physics; they are also easily reverse-engineered alien technology,

If you think of the Ark as a violation of "space and time," by following the rules above we can assume that through a person utterly violating the concept of "space" will result in a correction to balance it out - in the form of temporal anomalies where people may phase out of existence like Anderson. Take this in reverse, and you can see the dangers of utilizing time travel. Disrupt temporal stability enough, and the universe will correct the area by freezing time and resetting everything. Even on a small scale, minor "corrections" can destroy facilities, and disrupt the alignment of entire systems through the destruction of planets.

After a certain limit is reached, the Universe simply rejects all changes to the timeline, and the corrections will reset (though not always perfectly, and any parties involved with the tampering will be violently expelled from space-time. This expulsion varies from the affected person are at being killed outright and their corpse found among the explosions, or possibly removed from the timeline. This means that within a given area only so many distortions can occur before the limit is passed and the "reset button" is pushed.