Yes you should be able to access the "partner" websites from home.
If you are having trouble accessing a "partner" website, it may be that you have a pop-up block which is preventing access. [ If this is the case often you can click the pop-up blocker and allow your request to access the partner website. This knowledge article may be useful - https://www.familysearch.org/en/help/helpcenter/article/how-do-i-keep-pop-up-blockers-from-causing-issues ]
You do need an account with each of the "partner" websites. When you click on the partner logo you will be asked to sign into your account. Members of the Church of Jesus Christ of Latter-day Saints can sign up for a free membership with some of the "partner" websites. This knowledge article may be useful - https://www.familysearch.org/en/help/helpcenter/article/how-do-i-sign-up-for-access-to-partner-memberships .
The records which are available from home may differ from the records which are available through genealogical societies or local libraries which have "library" subscriptions rather than "personal" subscriptions. FamilySearch Family History Centres also have "library" subscriptions.
I hope you find this information useful.