Long-term injury reserve (LTIR) gives a team permission to spend above the cap, with the idea being that they might need that room to replace the injured player(s).
The exact amount that they are permitted to spend depends on whether they enter LTIR before or after opening night and on what their payroll and cap space looks like on the day they enter LTIR. But in general, the most important consideration is that they want their payroll to be as high as possible on the day they enter LTIR. Their spending limit for the entire time they are in LTIR will be related to their payroll on that date, so it is common to see a team add players to the roster to maximize their payroll on the day they go into LTIR - that locks in the favorable spending limit, which remains the limit even if they loan those players back to the AHL soon after.
LTIR can help a team make additions, but it does carry some disadvantages compared to being under the cap. Imagine a team starts the year with $1M of cap space and somehow gets through the whole year without a single transaction; they will end the year $1M under the cap. If they had players earn performance bonuses that year, the first $1M earned will be applied to the current year's cap - there will only be a performance bonus rollover to the next year if players earned more than that.
In contrast, imagine a team spent the whole year in LTIR, but $1M under their spending limit. Performance bonuses can only be applied to the current year if the team is under the cap; if they used LTIR to spend over the cap, it doesn't matter that they had LTIR room to spend even more - they will still have to roll all of their bonuses over into the next season, and will have less room to work with as a result.
Another advantage to being under the cap is the way cap space can accrue. Suppose that team with $1M of cap space gets 3/4 of the way through the season and decides they want to add a player. They are on pace to finish the year $1M under the cap, so they have enough room to take on a player with a $4M cap hit for the remaining 1/4 of the season. Their $1M of spending power has quadrupled!
In contrast, LTIR space does not normally accrue in the same way. The team that has been $1M under their spending limit all season still only has $1M to spend at the trade deadline.