I identified my observation after someone added a higher level-ID, so why is the observation stuck with the higher-level ID?

That's the way the Community Taxon system works: iNat chooses the taxon that more than 2/3 of the community agrees upon, and if that's impossible, it walks up the taxonomic tree and chooses a taxon everyone agrees with. For example, if I say it's Canis and you say it's Canis familiaris, 2/2 identifications agree it's in Canis but only 1/2 think it's Canis familiaris so iNat goes with Canis. You can see a more detailed explanation below.

If you don't like this and want your ID to take priority for your observation, just reject the community ID by clicking the "Reject" link under the community ID. You can also opt-out of community IDs entirely by editing your settings. 

You don't need to ask people to remove their higher-level ID, especially if it's accurate but not precise (eg Canis in the example above). This doesn't affect an observation's potential to reach Research Grade status, it just gives the observer control over what taxon the observation is associated with.

Community Taxon - A Detailed Explanation

The Community Taxon (or Community Identification) represents what taxon the iNaturalist community thinks is depicted in an observation. In general, we try to choose a taxon that more than 2/3 of the identifiers agree with. Sometimes this means choosing a higher level taxon that contains a number of disagreeing taxa (e.g. you think it's a kingsnake and I think it's rattlesnake, so iNat chooses suborder Serpentes which contains all snakes). The algorithm also slightly favors dissent, because we've found that dissenters are often correct.

A Research Grade observation must have (among other criteria) a Community Taxon. If an observation has only one identification, it won’t have a Community Taxon. All observations with at least one identification will also have an Observation Taxon. The Observation Taxon is the taxon we use when sharing observations with data partners, linking observations of the same taxon on the site, updating your life list, etc. 

In most cases the Observation Taxon will eventually be set to the Community Taxon, but sometimes they will differ especially before the community has settled on an identification. For example, if you think its a snake (suborder Serpentes) and I think it’s a kingsnake (genus Lampropeltis) the observation taxon will be at kingsnake (supported by my identification only) but the Community Taxon will be at Serpentes (supported by at least two identifications). If for some reason you don't agree with the Community Taxon, you can reject it on your own observations, which means that Observation Taxon will never be set to the Community Taxon (rather it will reflect your own identification). It also means your observation can only become Research Grade when the community agrees with you. If you don't like the whole idea of community taxa, you can opt out of them entirely by editing your settings.

The algorithm: for all identified taxa and the taxa that contain them (e.g. genus Homo contains Homo sapiens), score each as the ratio between the number of ‘agreements’ - cumulative IDs for that taxon over the sum of the cumulative IDs, ‘disagreements’ - the number of IDs that are completely different (i.e. IDs of taxa that do not contain the taxon being scored), and ‘ancestor disagreements’ - the number of more conservative IDs that disagree with the finer taxon. For the identified taxa that have a score over 2/3 and at least 2 identifications, choose the lowest ranked taxon.

You can see how identifications affect each observation's Community Taxon by going to that observation's page on the iNaturalist website and clicking on What's This? or About in the Community Taxon section.