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In Germany, there is an entry restriction in place for most of the EU citizens who were not German citizens or residents. According to the Federal Foreign Office of Germany, "This warning was lifted as of 15 June" (source).
It is not clear when exactly the borders open: after 15th June, meaning on June 16th, or from the beginning of 15th June, meaning 00:00:00 on 15th. I am asking for an Estonian citizen visiting Germany for 'nonessential' travel.
As I commented, the sentence you mention in the question isn't directly relevant, as it relates to the German Foreign Office's warnings against travel from Germany to other countries. Rules regarding travel to Germany are a matter for the Ministry of the Interior.
According to the German Ministry of the Interior
Bundesinnenminister Horst Seehofer hat das Bundeskabinett heute darüber unterrichtet, dass die wegen der Corona-Pandemie eingeführten Grenzkontrollen an unseren innereuropäischen Binnengrenzen mit Ablauf des 15. Juni 2020 enden. Das betrifft die Grenzen zu Österreich, Frankreich, der Schweiz, Italien und Dänemark. Die temporären Binnengrenzkontrollen im Luftverkehr bei Einreisen aus Spanien enden nach gemeinsamer Abstimmung in beiden Richtungen zum 21. Juni 2020.
Mit dieser Entscheidung entfällt generell bei Reisen aus anderen Staaten der EU nach Deutschland das Erfordernis eines triftigen Einreisegrundes. Dies gilt auch für Einreisen aus den Schengen-assoziierten Staaten Norwegen, Liechtenstein und Island sowie dem Vereinigten Königreich.
The statement is mostly about controls on the borders to Austria, France, Switzerland, Italy (presumably by air), and Denmark, but also covers general requirements for entry from other EU countries.
I've quoted the German because the English version omits some important details of the German statement. My translation of the above, with added emphasis on the most relevant sections:
Federal interior minister Horst Seehofer has today instructed the federal cabinet that the border controls on our European internal borders will end at the end of 15th June 2020. This affects the borders to Austria, France, Switzerland, Italy, and Denmark. The temporary interior border controls for air travel arrivals from Spain will end by joint agreement in both directions on 21st June 2020.
With this decision the requirement of good justification for travel from other EU countries to Germany is lifted. This also applies to travel from the Schengen associated states of Norway, Liechtenstein and Iceland, as well as the United Kingdom.
Specifically, the German text's use of the word "Ablauf" (~"expiration") makes it clear that the border controls are to be lifted at midnight at the end of the 15th.
As explained in the answer by Chris H, there is some confusion introduced due to the usage of the phrase "Ablauf des 15. Juni 2020 enden" in the official statement of the Germany Ministry of the Interior; since the word Ablauf roughly translates to "expiration" and hence implies the expiry of the day of June the 15th.
However, as for air travel, I have gotten a unanimous affirmative that Schengen citizens can enter Germany from (and including) June 15th 2020 without the need to prove "essential-travel" from the following authorities:
EDITS:
Disclaimer- the author of this answer has only functional knowledge of the German language.
I am myself not accepting this answer, since I can't prove in writing my confirmations. I guess, in the worst case, I'll wait until June 15th to see the outcome. Indeed, there were no border control measures in place since the beginning of the day today (i.e. June 15th since 00:00:00).
The decision was a bit weird because Switzerland and France are opening up from the moment June 15 begins (midnight between 14 and 15 of June) and it was supposed to be a concerted effort, but the wording of German announcement and the interpretation by some German border police were indeed indicating the border should be open on 16th. But nonetheless, it caused huge confusion, especially for Basel.
But Germany has now decided to practically lift border control at the same time as Switzerland in a last-minute decision (or clarification), that is, Germany will stop enforcing border controls at internal borders from the beginning of the day on June 15:
Schweizer können nun doch ab Montag nach Deutschland Am Montag gehen die Grenzen zwischen der Schweiz und Deutschland auf. Das war bis diese Woche allen klar. Doch dann sorgte ein Tweet der Bundespolizei Baden-Württemberg für Verwirrung: Die Grenzen würden erst in der Nacht von Montag auf Dienstag geöffnet, «mit Ablauf des 15. Juni», hiess es da.
Das hätte einige Probleme bereitet, so hätten zum Beispiel Deutsche zwar in die Schweiz einreisen, aber nicht mehr ausreisen können. Nun löste der Sprecher des deutschen Innenministers Seehofer am Samstag ebenfalls auf Twitter die Verwirrung auf: Die Grenzkontrollen würden in der Nacht auf Montag «praktisch» eingestellt.
Twitter from German Ministry of Interior spokeperson:
Mit der Aufhebung der #COVID__19 bedingten Binnengrenzkontrollen entfällt auch das Erfordernis des triftigen Einreisegrundes. Auf Anweisung von BM #Seehofer wird die #Bundespolizei die #Grenzkontrollen an den Binnengrenzen zum 14.6.2020, 24 Uhr, praktisch einstellen.
With the abolition of the COVID-19-related internal border controls, the requirement of a valid reason for entry is also abolished. On the instructions of Minister Seehofer (German Minister of Interior), the Federal Police will practically lift border controls at the internal borders on 14.6.2020 at 24:00.
