I noticed my colleagues use this phrase a lot during meetings and discussions in all kinds of situations, usually as an adjoint to what they`ve just said.
Without context, the phrase translates to »at that point«. An example could be
Ich stimme zu, dass wir die Aufgaben neu verteilen sollten, an der Stelle (zumindest).
I agree that we should reassign the tasks, at that point (at least).
Here the speaker relativizes or restricts what they said before. Without the appendix, they would claim ask to reassign tasks tout court. Adding »an der Stelle« makes clear that they want to reassign tasks only with respect to what has been previously discussed. In such a context, one puts stress on »der«.
Another example is
Ich habe einen Fehler entdeckt, (und zwar) an der Stelle.
I found a mistake, there.
This would be accompanied by a pointing gesture to show where the mistake is.
That said, I find both of these examples somewhat unnatural, and would be surprised to hear them a lot. Perhaps what you hear was a similar-sounding phrase such as »an deiner Stelle« (if I were you)?
Edit: An example where the phrase is not used as an adjoint is
An der Stelle ist interessant, dass ...
At that point it's interesting to note that ...
To me, that would be a natural way to make some side comment / remark, similar to saying »by the way«.
It is most likely used as a filler. Depending on context it might also indicate either pinpointing something (e.g. pointing at a figure on a powerpoint slide. "an dieser Stelle sieht man, dass die Zahl 10 größer ist als die Zahl 5") or setting a temporary/causal context ("Zunächst nimmt Wert X ab, an dieser Stelle sieht man, dass er nun geringer ist, danach nimmt Wert X wieder zu").
An english synonym could be: "As you can see here", "For instance", "lookie here"
or even something as plain as "per se".
An dieser Stelle you can see, that the usage of the term is pretty context related and can be safely ignored in most cases.