It means that arcane connections have expiration dates similar to food from the supermarket.
So shed skin from a person will remain an arcane connection to that person for a few hours while the mummified arm of said person would remain an arcane connection for decades. Other things fall on a spectrum somewhere in between there two extremes.
So yes a persons shed skin only lasts hours but an invested item made by them lasts for years.
In essence what it boils down to is that arcane connections are magical connections between things and things that derive from the first things. Some things have a stronger magical connection to other things and some things have weaker magical connections. Now I realize that I said thing way too much in the previous 2 sentences and I think it is better explained by examples, much as is done in the core book in fact.
Imagine a person. That persons bodily parts are naturally linked to the person, as mentioned above, both shed skin and mummified arms are things that were previously parts of the person, and for that reason they are magically connected to the person (but not vice versa). But not all connections are equal, shed skin is not a very big part of the person, metaphysically (and physically) speaking, so its connection to the person is weaker and thus expires quicker,, opposite goes for body parts.
Part of the reason for the expiration of small insignificant things like shed skin, air from a room etc, is that these things by virtue of being only weakly connected can easily become connections to a new thing instead, and thus loose their original connection. Bottled air from a room quickly become a connection to the bottle instead, shed skin becomes ordinary dust and thus becomes a connection to the room where the dust settles etc.
It is useful to consider how central the connection was to the thing it connects to before they were separated. Is dust or air a central part of what a room is? not really. A persons arm however is likely to form a significant part of their identity, as well as making up a substantial portion of their actual mass so the arm is a strong connection.
It is worth noting here that indefinite duration connections are AFAIK always derived from some magic that sustains their indefiniteness. Magi and their familairs, magi and their talismans and fixed arcane connections.
Since the duration scale for arcane connections maps to strength of association between connection and connected the scale is nicely useful for gauging how much of a bonus the connection ought to provide, which is what you see in the list with the column named "bonus to multiplier". Note that the bonus to multiplier is IMO not very easy to understand but as far as I do understand it what you do is take all of your "bonuses to multiplier" add them together and add 1, then you multiply the penetration of your spell by the number you got. Meaning that an arcane connection makes it much easier to cast spells on targets with magic resistance.
Arcane connection durations has no direct bearing on the duration of the spell. It imposes a limit on how long you can wait before casting the spell using the arcane connection. If you try to use an arcane connection that has expired your spell will fail since it is not an arcane connection anymore, and you thus dont meet the criterium of having an arcane connection.
Yes this does make selling items dangerous. I think it is quite likely that buy invested items comes with a very long shipping duration, but then again invested items are expensive to make and probably sold rarely and only to order. It makes items created a long time ago valuable for the fact that they can be safely sold. As to why sellers of enchanted items ever offend other magi I think the answer is the usual answers. Most people enchanters or no dont often set out to offend others it just sort of happens to them by accident or pride.
Depending on your stance the arcane connections can also be reduced in strength with PeVi spells.