Everything coming out of the White House these days is begging for some kind of independent investigation. But what does that even mean? If I read that X politician or troubling scenario is being independently investigated, what makes them "independent"? Assume that this supposed independence is legitimate and not some publicity farce.
The key for an independent investigation is that the investigators must have no official ties to the people they are investigating. This is commonly used if there is a suspicious of systemic issues at an institution: if everything at an institution is broken, then the investigation will be too. Therefore in cases like this an outside group will be brought in and by virtue of being from the outside and reporting outside the system it will not be influenced by internal politics.
A recent example of this is the accusations of Uber's systemic culture of sexual harassment. Uber has a legal and HR team that can do this investigation, but since they are part of the accused any such investigation would be immediately tainted. As such, Uber brought in Eric Holder (former AG of the US) to do the investigation. He Holder is outside of Uber and does not report to them, therefore he should not be influenced by any internal issues.
With all that being said, we're all human and influences can reach far (especially in government). The true independence of any investigation can always be called into question, so its ultimately up to you to judge if you believe its objectivity.