You become invisible at the same time that an illusory double of you appears where you are standing. The double lasts for the duration, but the invisibility ends if you attack or cast a spell.
You can use your action to move your illusory double up to twice your speed and make it gesture, speak, and behave in whatever way you choose.
You can see through its eyes and hear through its ears as if you were located where it is. On each of your turns as a bonus action, you can switch from using its senses to using your own, or back again. While you are using its senses, you are blinded and deafened in regard to your own surroundings.
Illusion spells are not tangible in general.
The mislead spell creates an "illusory double," so we can look at the description of the Illusion school of magic for more insight (see Casting a Spell in the SRD, emphasis mine):
Illusion spells deceive the senses or minds of others. They cause people to see things that are not there, to miss things that are there, to hear phantom noises, or to remember things that never happened.
Since the illusion created by mislead is not really there, it has no physical presence. It could not be touched or bumped into, so it is literally not tangible.
Some illusion spells might create illusions that are tangible, but if they do then we would expect them to explicitly state that they are exceptions to their school of magic. Since mislead does not state any exception, we have no reason to think that it is one. In addition, if it was possible to physically interact with the illusory double, we would expect that the designers would call it an object or a creature, but in the absence of either of those words we must simply assume it to be an effect that behaves exactly as described.
(This is under the interpretation of the word "tangible" meaning "material or substantial." That is, tangibility is an inherent property of a thing regardless of who is or is not perceiving it.)
There is no reason to believe that it doesn't
"Illusion spells deceive the senses or minds of others." They are not limited to any particular senses unless the spell says they are. This particular spell does not limit the illusion in any way so there is no reason to believe that it cannot be seen, heard, smelled, touched and even tasted if you are so inclined.