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The earthquake destroyed the external power supply of the nuclear reactor. That is one of the most serious accidents for a nuclear power plant, and accordingly, a “plant black out” receives a lot of attention when designing backup systems. The power is needed to keep the coolant pumps working. Since the power plant had been shut down, it cannot produce any electricity by itself any more.

The term is "station blackout" when both the off-site and on-site AC sources are unavailable. The off-site system was not destroyed, just disconnected. TEPCO announced that off-site power has been restored. Plants uses off-site power in various ways, depending on licensing and design. A common way is for the electric equipment to normally run on off-site power because it is more reliable. The plant generates power and feeds it into the grid via a transformer. Other transformers connect the off-site grid to the in-plant equipment. Some plants cross connect these transformers, some do not. There is no English language detail for Fukushima Daiichi No. 1 in this context. Marine diesel engines run on-site generators to produce power for safety-related equipment when off-site AC is unavailable. Fukushima Daiichi No. 1 is an older design and approaches the "station blackout problem" with a unique set of systems. Units 2 through 6 have a RCIC system that is able to pump water using the steam being formed in the reactor by decay heat. Units 2 through 6 also have a HPCI that also uses a steam driven pump. Both of these pumps can only work when pressure in the reactor is low enough to be overcome by the pump head. Safety relief valves and automatic depressurization valves vent steam into the torus so that HPCI and RCIC can pump liquid water in. HPCI and RCIC control needs DC power and can (with some imagination) be run without those controls.

The diesel generators were destroyed by the tsunami.

This is pure speculation. The diesels are housed in the same earthquake resistant, flood resistant building as the reactor equipment (details vary at each plant site, but the government regulations related to the emergency diesel generators are the same regardless of which plant you go to). It is possible that the cooling water for the diesels was affected by the tsunami and running the diesel with that water caused a problem; however, diesels are used in these applications specifically because they are rugged and durable and able to take some degree of abuse. The article above states with unusual certainty that the tsunami destroyed the engines -- I find the term destroyed a bit over the top and hard to believe.