Germany

Some states in Germany have a heightened risk of SARS-CoV-2 infection and have been labelled by the Robert Koch Institute (RKI) as high-risk areas. If a country or region is declared a high-risk area, entry is subject to the quarantine regulations for the individual states. The governments of the states and the federal government agreed on a template decree and the individual states essentially issued the same wording. Upon entry, travellers must continue on to their destination immediately and enter home quarantine until they present a negative coronavirus test and report to the relevant authority. Travellers are also required to have themselves tested for coronavirus.  Information on the individual federal states can be found in the decrees they each issued. For travellers from the Czech Republic to Germany, the most relevant federal states will be Bavaria and Saxony, which share a border with the Czech Republic. 

Due to the rising number of infections in the Czech Republic with the R number exceeding 50 cases per 100,000 residents, on the evening of September 9, 2020 Prague was declared a high-risk area.

Bavaria
According to the decree on quarantine measures to fight against coronavirus (Einreise-Quarantäneverordnung – EQV), travellers entering Bavaria must remain in mandatory home quarantine upon arriving or returning if they spent time in Prague in the 14 days prior to entering Bavaria. To avoid this quarantine period, people must have a negative coronavirus test in German or English and present it immediately upon request. The authorities must also be informed immediately. The test must be made no more than 48 hours before entering Germany or immediately after entering Bavaria. Those entering Bavaria from a high-risk area can have a coronavirus test done within 72 hours free of charge.

The obligation to present a negative test and enter quarantine does not apply to cross-border workers (pendlers) regardless of where they stay in another country as well as persons transporting goods and passengers across borders and airline, ship, railway and bus employees. This obligation also does not apply to people who spent less than 48 hours abroad and people travelling to Bavaria for absolutely necessary medical or work reasons or for reasons that cannot be delayed. People travelling to Germany for social reasons such as joint custody of a child, visiting a domestic partner or providing assistance to vulnerable individuals, are also exempt. Essential workers (people whose activities are essential to maintain public safety, order and similar activities listed in the decree) are also exempt. Other exemptions may 
be granted in justified cases. The exemptions apply only as long as travellers do not show symptoms of illness.

The regulation was extended through September 18 and may be reworked in the near future. Special regulations also apply to individual districts in Bavaria, such as Rosenheim and Landshut, due to their own rising infection numbers. 

The current decree can be found here: 
https://www.gesetze-bayern.de/Content/Document/BayEQV/true?AspxAutoDetectCookieSupport=1 

Saxony

According to the coronavirus decree in Saxony, people travelling and returning to Saxony must adhere to a home quarantine period of 14 days. This also applies to people who first entered a different federal state. Transit passengers and passengers who can show a negative coronavirus test in German or English are exempt. The test must not be more than 48 hours old. In the first 72 hours after entering Saxony the test is free of charge. 

Also exempt are people who work in transporting goods and passengers, people who work for airline, ship, railway and bus companies, and members of the armed forces and police. Another exemption applies to people who travel to Saxony for work for at least three weeks and who have gone through the group health measures and avoided contact with others during the first 14 days after their arrival (seasonal workers). In justified cases the relevant health authority can 
decide to grant other exemptions. 

UPDATE: From 10 September 2020 the Saxony quarantine regulations have changed. Now cross-border workers (pendlers) and people travelling to Saxony within a limited time period for absolutely necessary work, medical or social reasons are also excluded from the obligation to present a negative test or enter quarantine. Essential workers (people whose activities are essential for the care of diplomatic relations and similar activities listed in the decree) are also 
exempt.

The exemptions apply only as long as travellers do not show symptoms of illness.

The current decree can be found here: 
https://www.revosax.sachsen.de/vorschrift/18739-Saechsische-Corona-Quarantaene-Verordnung#p3 

and the changes can be found here: 
https://www.coronavirus.sachsen.de/download/SMS-DritteAenderungsVO-Quarantaene-2020-09-10.pdf 

Other information on Austria
According to official sources, Austria is currently not planning to close its borders or warn against travelling to the Czech Republic. According to the Austrian Foreign Minister, the epidemic situation in the Czech Republic will continue to be closely monitored and reassessed as needed.

Austrian rules currently state that anyone entering from areas that have been declared high-risk must either present a doctor’s note with a negative PCR test (no more than 72 hours old) or must enter a 10-day home quarantine. The quarantine period may end early, however, if a PCR test performed during that time comes back negative. Exemptions from these restrictions apply to pendlers (with confirmation from the employer), for serious family reasons and for transport of goods and passengers.

For more information, please see: 
https://www.sozialministerium.at/Informationen-zum-Coronavirus/Coronavirus---Haeufig-gestellte-Fragen/FAQ--Reisen-und-Tourismus.html 
https://www.ris.bka.gv.at/GeltendeFassung.wxe?Abfrage=Bundesnormen&Gesetzesnummer=20011194