How can we interpret: "Eat from what is near you!"?
Lets start from a common sense perspective:
In this hadith the Prophet seems to teach his stepson and nephew -by lactation- (as 'Omar ibn abi Salamah is the son of Umm Salamah) some table manners, after observing him. 'Omar even explained a few of his mistakes:
"... as my hand used to roam about in the dish..." (sahih Muslim and sahih al-Bukhari)
- So the order or recommendation might be understood as, don't mess around with your hand while eating, it is better to eat from what is in front or near you!
- That means if they eat from one single plate, don't go and reach out to eat from the part of the plate which is next to somebody else. Or just ask for permission before doing so, as we usually do if we like somebody to hand over a plate with lets say olives or salad etc. if it is not close to us.
- If they eat from different plates and have different dishes on the table it is not recommended to eat from somebody else's plate unless one has permission or as stated above asking people who are closer to give it to you!
- Finally picking up a piece of food here and there is not really hygienic, especially if you are not the only person who eats from those plates etc.!
So it could include both meanings however the meaning what is on your or the plate is the most obviouse if we have in mind that many Muslim meals are presented in a single "big" plate. And it is recommended to eat using three fingers (see for example in sahih Muslim). So using hands instead of cutlery was usual and some people still insist on that as it is "sunnah".
Interpretation based on hadith commentaries
The hadith of ibn Abi Salamah is compiled in both sahihs, sunan ibn Majah and Jami' at-Tirmidhi. I'll quote some interpretations from the commentaries of both sahihs:
An-Nawawi in his commentary on Muslim explained "my hand used to roam about" as follows:
( تطيش ) ... ؛ أي تتحرك وتمتد إلى نواحي الصحفة ، ولا تقتصر على موضع واحد
--- (My own translation take it carefully) ---
means that his hand used to move a lot and he stretched it to many directions of the plate and didn't keep it in a single "area".
He also distinguished between as-Safha الصحفة and al-Qas'a القصعة (both are synonyms for plates), the difference lies in the size, so as-Safha (as stated in the hadith) is smaller, but is big enough to feed about five individuals while al-Qas'a is big enough to feed ten of them according to al-Kissaa'i (Quran reciter and linguist).
He added that this hadith shows three good manners (sunnan, plural of sunnah) of eating (table manners): tasmiya or naming (saying bismillah), eating with the right hand (which were already explained due to earlier ahadith in the chapter) and eating from what is close to somebody: Because eating from in front of somebody else is considered as a bad company and no sense of honor especially when it comes to soups and sauces and likewise food, as other people may detest such an act (as it seems for hygienic reasons), while it was reported that stretching out hands for "non-liquid" food like dates is allowed. But it should be considered that the general "prohibition" holds as long as there's no clear evidence!
Ibn Hajar al-'Asqalani in his fath al-Barry comment on sahih al-Bukahri added explaining the expression of 'Omar ibn abi Salamah "my hand used to roam about" by an other narration of him in sahih al-Bukhari. he also quoted that most scholar of the shafi'i school consider "eating from what is in front of you" as a recommendation -mandoob- (he quoted among them al-Ghazaly, an-Nawawi, al-Baydawi, Taj ud-Diyn as-Subki ) while a-Shafi'i him self said it is wajib -a must!
And he explained "eat from what is in front of you" saying: If the meal is from one kind, as if in that case you toke from an others "portion" that would be a kind of violation of that person beside the fact that the meal portion might be touched with some "filth" (Note that this indicates the hygienic aspect of that act, this was also somewhat mentioned by an-Nawawi) and it is beside this a bad manner without any use. But if the kinds of food or meal (maybe also the plates) where different than it is allowed to eat from elsewhere.
If everybody has an own plate one is asked to eat from it (the whole plate not only what is near to you!) and shouldn't eat from somebody else's plates (for hygienic reasons).
If there are different dishes/meals/kinds of food it is allowed to take from here and there however it would be better not to disturb other people.
If theres a single plate you again are asked to it from what is in front of you (for hygienic reasons).
And here a fatwa on eating etiquette quoting this hadith and others roughly.