Pretty much if you're including an unnecessary word like "io" here, it's for emphasis or clarification, and if you change the structure ("Io vado" to "Vado io,") you are further emphasizing the "I" and probably pronouncing the "io" with extra emphasis.
By clarification, I mean if want to clarify He vs. She, or "(Io) sono" vs. "(Loro) sono," where the subject would be ambiguous.
The link provided by MrMacbeth is to the Italian to English course, while we are in/at the English to Italian course. The comments there suggest that the Italians are just as confused about the meaning of "al tuo posto" as we are - and they are having difficulty deciding the same question that we are, except they seem to be wondering, in addition to meaning "instead of you", it also means "to your place".
I did come across an Italian phrase, "al posto di te" which means the same thing as "invece di te", which translated as "instead of you". That indicates to me that both translations are quite correct, with two very different meanings:
1. I go to your place - your house, your apartment, your place in line, etc.
2. I go instead of you, in your place, doing what you would have been doing.
I'm satisfied, at least for now, that this exercise sentence can mean both things, and it would (Surpise!!!) take more context to determine which one.
It's the same as saying, "I go to your house" vado alla sua casa
Both "in your place" and "to your place" are accepted. Apparently, "posto" can mean both things that it means in English:
1. Your spot, location, place
2. Instead of you. I've encountered al posto di te as an equivalent to invece di te, "instead of you" = "in your place".
I went to the Italian-English course, and they are discussing the same issue, except wondering whether the English has both meanings in the way the Italian does. Well, except, most of the people there seem just as uncertain as the people here, me included, about the certainty that it means both.
Yes, the following examples from Reverso Context suggest that it can mean "in your place" in both an abstract sense and in a more physical sense (as in a seat in an airplane)
1) I understand "io" is included for emphasis, but why doesn't it go before "vado"?
2) Is posto used here figuratively? As in "I'll take your place in the upcoming race" instead of "I'm coming to your place now (e.g. apartment)".
3) Why is "al" used? Doesn't it mean "at" in english?
As I understand it . . .
1) Italian allows you to add/move the subject pronoun around for emphasis. E.g., Vado = least emphasis, io vado = greater emphasis, vado io = greatest emphasis.
2) Posto here does mean something like your place in a race. However, it is not figurative, but quite specific. Hence the "al" a+il. Think of the "il" part of "al" as being there because of "posto". If you wanted to say I'm going to your place (as in apartment or home) it would be "Vado a casa tua".
3) Italian is a VERY idiomatic language. You'll find that words like "a" are introduced to you as meaning "to" or "at" but then can be used to mean "in" or "on" or "of", etc. It's one of the challenges of the language.
Italian in in your phrase I believe means what it appears to say: "in" - that is, in the physical place, a very limited space. It doesn't make much sense to consider trying to go in somewhere, when you're limited to one small location.
al on the other hand has a kind of multiple simultaneous meaning which can partake of "at, to, as, & in" all at the same time. By being kind of fuzzy in meaning, the grammar allows for multiple significations to attach, so that you as going to someplace while acting as someone in their role with their rights and obligations.
Maybe think of the dessert, "Pie a la mode. It's not Pie in la mode or Pie di la mode* - and there's a foreign-language, grammatical reason for that which is extremely difficult to explain rationally.
I have a guideline - if you don't know, and there's nothing about the context or sentence structure which suggests otherwise, use "a" rather than "in" or "di" - you're more likely to get it right. It works in French and Spanish, and it seems to work in Italian.
An occasion when someone would use this could be after long deliberation, the speaker says, “I know! Rather than …, let me go in your place,” (or “let me go to your place,” or “Why don’t I go in your place.”) So, it’s just a grammatical construct for placing the emphasis on “io.”
If you just said “Vado al tuo posto,” which is completely correct and common, I would interpret that to be just a regular statement, rather than an emphatic statement (e.g., a new idea, a possible solution to a problem, an unexpected statement, etc.).
I hope that makes sense.
I've just now asked my friend who's from Venice. She says it means doing something for someone; for instance if someone is not able to kerp an appointment and someone else says 'I'll go for you' or 'I'll take your place'. She gave me some altetnative ways to say 'I go to your place' but too much to put here...