When you swear to do something, you're saying you will do it. This can be interpreted as a religously backed oath or as a sincere promise. [1] Either way, quakers object because they believe they're supposed to tell the truth at all times, and so they "do not swear, but we 'affirm' that we are being honest, as always". [2] The word of a quaker, however, has lost a lot of its reputation. People no longer trust arbitrary statements by most quakers more than those same statements made by non-quakers. This is because quakers have, over time, become less serious about telling the truth. Quakers have become much more integrated into society than they used to be, and now mostly have accepted the prevailing norms on truth telling. Which isn't to say that quakers lie all the time any more than most people lie, but that requesting promises not to lie from quakers makes sense.

Further, you're being asked to tell the "whole truth". Quakers who take not lying very seriously have traditionally still occasionally used not telling the whole truth as a way around that strictness. There are stories about people giving intentionally misleading but not technically false responses to questions like "why would you think that?" and non quakers warning each other to force a quaker into giving a straight answer and not to be misdirected by apparent denials that aren't actually making any claim. So even if you do believe that you should be always telling the truth and nothing but the truth, I'm not sure about the whole truth.

Finally, just because you were already going to do something doesn't mean you can't promise to do it. Quakers traditionally promise to be loving and faithful to their spouse. Does this imply that they were not already going to be loving and faithful? That if the marriage vows did not contain that promise, those actions would not still be part of marriage? Similarly, one who already belives they must always tell the truth, the whole truth, and nothing but the truth can still promise the court that they will do so.

[1] The word "swear" doesn't have to indicate that the following statement is religiously backed. The first definition is: "make a solemn statement or promise undertaking to do something or affirming that something is the case".

[2] This is ohio yearly meeting's phrasing