The Trade Federation knows Sidious is a Sith Lord. For example:
Nute: This is getting out of hand...now there are two of them.
Rune: We should not have made this bargain. What will happen when the Jedi
become aware of these Sith Lords?
Star Wars: Episode I The Phantom Menace script
They also know he has influence over the Senate, as implied by your quote and others. They almost certainly did not know Sidious was Palpatine, the senator from Naboo (as it would make no sense for a senator to order an invasion of his own home planet). They almost certainly did not know (or even suspect) Sidious was a senator, either. Sidious need not be a senator in order to have such influence. Recall what Dooku told Obi-Wan in Episode II about Sidious and the Senate:
Count Dooku: What if I told you that the Republic was now under the
control of a Dark Lord of the Sith?
Obi-Wan: No, that's not possible. The Jedi would be aware of it.
Count Dooku: The dark side of the Force has clouded their vision, my friend. Hundreds of senators are now under the influence of a Sith Lord called Darth Sidious.
Star Wars: Episode II Attack of the Clones, transcribed from a Youtube video of the scene
Dooku claims that Sidious has influence over the Senate (just as the Trade Federation knows) but he implies that Sidious' influence is based on his control of "hundreds of senators", not that Sidious is a senator (or Supreme Chancellor). In fact, it makes more sense to assume that Sidious privately controls hundreds of senators (e.g. by bribes, blackmail, etc.) than that Sidious is himself a senator: hundreds of senators are more powerful than a single senator (Sidious himself).
The Trade Federation was probably given the same implication that Sidious was (a) a Sith Lord who (b) had influence over many senators. There's no need or reason to tell the Trade Federation that Sidious is a senator, much less the Senator of Naboo.
The Trade Federation followed Sidious' instructions throughout the Naboo blockade and Clone Wars because Sidious got results for them in the Senate (e.g. by preventing the Senate from acting against the Trade Federation's blockade). They don't know how he got those results, but they don't need to.
As for why the Trade Federation followed Sidious' instructions before they took the risk of blockading Naboo, canon doesn't explain much. The Trade Federation probably just trusted Sidious' power as a Sith Lord. Legends has more background: the novel Cloak of Deception includes an incident in which Sidious engineered Nute Gunray's rise to power as Trade Federation viceroy. Interestingly, in Cloak of Deception Sidious used his alter ego Palpatine to propose a tax on Trade Federation trade routes in exchange for permitting the Trade Federation to build up their armed forces. The proposed tax benefited Palpatine/Sidious by giving the Trade Federation the means and motive to oppose the Senate and its taxes, and further ensured the Trade Federation would not connect Sidious' two identities (since Palpatine the Senator of Naboo was an enemy of the Trade Federation). As an even further bonus, the Trade Federation's anger at Senator Palpatine led them to attack Naboo, which ultimately gained sympathy for Palpatine and facilitated his rise to power as Supreme Chancellor. This was a win all around for Palpatine/Sidious and yet another way in which he proved himself a political genius able to manipulate both sides.