A Tag Management System, also known as a TMS is a web based tool, that cuts down on implementation time used for analytics/marketing tools. The amount of time the develop can be saved actually grows based on the number of tools you use. Each tool has different data requirements and even could have the same data requirements but in different formats. Meaning that product "A" could require product revenue sent over including the "$", and product "B" could want the same piece of information without the "$". TMS's allow the creation of a data layer's which then can be utilized to send the revenue using the same reference in the data layer, except product "A" will have the "$" in front of the token.
What is a data layer?
How does this save me time?
Making changing to the TMS/Data Layer do not require releases of code. So depending on your organizations time to market on releases this is where some time savings come from, besides the shared data layer.
What about making sure I don't introduce bugs?
Most TMS tools allow you to enable to changes only in your lower environments before requiring approval to be promoted to production. This will give you and your analytics team the ability to test all changes/additions and then publish this changes once the appropriate parties sign off. Typically this can be weeks faster than waiting for code promotions to get into "the wild".