This article has a recording of the Jewel Voice Broadcast, in addition to the original transcript and the modern Japanese translation of the speech. As you can see, the audio quality is far from good, but let's set that aside.

Generally, the Japanese used in 1945 is still understandable in the same way that 1945 English is still understandable today, with some differences: 1) the increase in use of loanwords (外来語 gairaigo and 和製英語 wasei eigo) - this is because in wartime Japan, the use of 敵性語 tekiseigo - the enemy's language - was generally discouraged, and 2) the consolidation of the ゑ and ゐ kana into え and い starting in 1946). Here's a pilot training video from 1945; the contents of the video are overall still understandable to modern viewers.

However, the Emperor uses 漢語 (kan-go) in his speech, a form of Japanese that heavily uses Chinese loanwords and old/outdated conjugations. For example, in the speech you can see very uncommon words like 加之 shikanominarazu (modern: それにくわえて sore ni kuwaete) or 康寧 kounei (modern: 平和 heiwa, 安寧 an'nei), as well as conjugations like セシメタリ seshimetari (modern: ～させる; the translator uses 指示した) or the ム form (開カムト欲ス hirakamu to hosu; the translation renders this as 開いていきたい hiraite ikitai). I'd caution in comparing the Emperor's Japanese to any form of English, but for comparison's sake, it would be like reading the Lord's prayer somewhere around Middle English to early modern English; if you stare hard enough you might be able to understand it, but it will take a whole lot of effort.

I hope this answers your question!

Despite the popular belief, the style used in the speech is not the language spoken in the court at the time, but an artificial language known as kanbun kundoku-tai. Hirohito was born and lived in an isolated palace, but went out to an elementary school for the peerage, and regularly attended social events and meetings. His language was no different from that of Japanese upper class, and the style of that speech was alien to the emperor as well. If you listen to the audio file carefully, you can see he himself was struggling to read the text.

The kanbun kundoku-tai is, roughly speaking, classical (Heian) Japanese so constructed as to look as if automatically translated from classical Chinese. It was originally developed in the Heian period as a method of translating Chinese texts without losing original meanings, but soon became a formal style of constructing literary Japanese sentences, not necessarily as works of translation from Chinese texts.

Kanbun kundoku-tai sentences follow the Japanese word order of Subject-object-verb, but tend to contain a lot of Chinese words not really “borrowed” into spoken Japanese, and use expressions unique to the style, which were specially invented for the style to keep as many original characters as possible. The conjugation follows the classical Japanese grammar, which is vastly different from that of modern Japanese.

Classical Japanese texts are always read in modern pronunciation, which sounds like good news but actually isn’t. As Japanese phonology is far simpler than that of Chinese, hundreds of different Chinese characters happen to be pronounced the same in Japanese, and that makes an endless list of homophones. It is not a problem as long as it is used as a written language, but caused great confusion on this specific occasion.

The first sentence

朕深ク世界ノ大勢ト帝國ノ現狀トニ鑑ミ、非常ノ措置ヲ以テ時局ヲ收拾セムト欲シ、兹ニ忠良ナル爾臣民ニ告グ。

is constructed with the following pseudo-Chinese sentence in mind.

朕深鑑世界大勢與帝國現狀、欲以非常措置収拾時局、兹告忠良爾臣民。

I (the Emperor), deeply pondering the general trend of the world and the actual conditions of the empire, want to settle the present situation with an extraordinary measure, and here talk to you, my good and loyal subjects.

The first word 朕 is the first noun exclusively used by the emperor. 爾 is an archaic from of the second noun, and the equivalent of English “thou”.

The preposition 以 “with”, the modal verb 欲 “want to” and the demonstrative adverb 茲 “here” are translated into ヲ以テ, ト欲シand 兹ニ without replacing the original characters. These Japanese expressions are only used in the kanbun kundoku-tai, and make the sentence highly formal.

All the verbs and the adjectives are conjugated according to the classical Japanese grammar. In modern Japanese, 収拾セム shūshū-semu would be 収拾シヤウ shūshū-shiyō, 忠良ナル chūryō-naru would be 忠良ナ chūryō-na and 告グ tsugu would be 告ゲル tsugeru.

The noun 大勢 taisei must have been misunderstood by the majority of the audience. In Japanese, 大勢 dàshì “general trend” is pronounced exactly the same as 退勢 tuìshì “declining trend”, 態勢 tàishì “attitude”, 體勢 tǐshì “posture” , 體制 tǐzhì “organization”, 大政 dàzhèng “Imperial rule”, 大成 dàchéng “great achievement”, 耐性 nàixìng “tolerance level”, 泰西 tàixī “the West”, etc. I imagine many of the audience interpreted the word as 體勢 “posture” or 體制 “organization”, both occasionally used in spoken Japanese, while 大勢 “general trend” is not.

The second sentence is the core of the speech which mentions Japanese surrender to the Allied Nations.

朕ハ帝國政府ヲシテ米英支蘇四國ニ對シ其ノ共同宣言ヲ受諾スル旨通告セシメタリ

I (the Emperor) have made the government of the empire notify the four countries, the United States, Great Britain, China and the Soviet Union, to the effect that it accepts their joint declaration.

[Object] ヲシテ [Verb] セシメタリ ([O] oshite [V] seshimetari), is the perfect form of the causative expression only used in kanbun kundoku-tai. The modern equivalent is [O] ニ [V] サセタ ([O] ni [V] saseta). ヲシテ and セシメタリ are placed so far apart that it is hard to grasp the structure of the sentence.

In the Japanese text, the Allied Nations are only described in the acronym 米英支蘇 Bei Ei Shi So. This is a result of the effort to avoid using non-Chinese loanwords like “America” and “Soviet”. Hirohito pronounces this part very slowly so that the audience can catch the meaning, but this expression makes the part needlessly difficult to understand.

And finally, you would miss the whole point of the sentence unless you are familiar with the content of “their joint declaration”.

Followup: The Showa Emperor made the infamous understatement in the broadcast that the situation had developed "not necessarily to Japan's advantage". Is this really a fair rendering of what he expressed, or a stereotyping slant by translators?

What factors lead to an emperor who spoke such an anachronistic version of his language that understanding him took effort? Was he surrounded by courtiers who interpreted his words? Did the generals and government officials who conferred with him understand him?